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CHE 625
TOPIC 3.2 : Diffusion and
ReactionRef: Fogler (Chapter 12) 4th ed.
Pg 813 onwards
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SUBTOPICS
Diffusion and Reaction in Spherical Catalyst PelletsEffectiveness factorEstimation of diffusion and Reaction limited regimesPore diffusion Resistance and Surface kineticsPorous catalyst particles
Gas liquid reaction on solid catalyst reaction and itsapplication
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CHEE 32322.3
Internal Mass Transfer in Porous Catalysts
Previously, we have examined the potential influence of external mass transferon the rate of heterogeneous reactions.However, where active sites are accessible within the particle, internal masstransfer (molecular diffusion) has a tremendous influence on the rate ofreaction within the catalyst. Numerous examples exist:
Encapsulated or entrapped enzymesMicroporous catalysts for catalytic cracking (zeolites)
The diffusion rate of reactants and products within the particle oftendetermines the rate at which a microporous catalyst functions.
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CHE625
Concentration Gradients of Diffusing Reactants and ProductsIn a Uniform Catalyst Particle
Spherical catalyst particle with diffusingreactant. C As is the concentration ofreactant at the particle periphery.
CAs
* Concentration at R (pore mouth) should behigher as compared to concentration at anypoint inside the pore
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22.5
Intraparticle Mass Transport
A simple but conceptually useful treatment of intra-particlemass transfer describes the diffusion of a reactant within auniformcatalyst particle.
Assumptions:Uniform catalytic activity throughout the spherical particleUniform properties of solid
Irreversible, first-order kinetics; rate = k C A
Given that mass transfer occurs by molecular diffusion, ananalytical expression for the transport and consumption ofour reactant can be written.
CAs
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Pores are tortuous and have varying cross-sectional areaNeed to define an effective diffusion coefficient in radialdirectionEffective Diffusivity (De) is a measure of diffusivity thataccounts for the following:
Not all area normal to flux direction is available for molecules todiffuse in a porous particle ( P)
Diffusion paths are tortuous ( )Pore cross-sections vary ( )Internal void fraction, s = P
Go thru’ Example 12 -1 pg 815 (Fogler)
Diffusion/Rxn in Porous Catalysts
~DD P A
e
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1. Perform Shell Balance in the radial direction and outward(increasing r) - in terms of molar flux times area
2. Write the expression for WAr (for EMCD and dilute
concentrations )3. Write the corresponding rate law (relate –r A, -r A’ and -r A”) and
relate with shell balance4. Set boundary conditions for C A (based on position r )
5. Write the equation in dimensionless form to obtain Φ (Thielemodulus) in terms of two dimensionless quantities ψ and λ
6. Obtain W expression with dimensionless quantities included7. Determine concentration profile for the spherical cat. pellet.
STEPS IN INTERNAL DIFFUSION &REACTION
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Diffusion/Rxn in Porous Catalysts
steady state mass balance
rate in at r
r
2 Ar r 4W
rate out at r + r
r r
2 Ar r 4W
rate of generation within shell
c
mass catalystrate reaction
volume shellmass catalyst
volume shell
rr + r
R
r r 4 2m
'
Ar
0r r 4r
r 4Wr 4W
2mC
' A
r r
2 Ar r
2 Ar
0r r dr
r Wd 2C
' A
2 Ar
B A cat
Step #1 : Perform Shell balance in radial direction
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Diffusion/Rxn in Porous Catalysts
0r r dr
r Wd 2C
' A
2 Ar
dr dC
DW Ae Ar
0r r r dr dCDdr d 2C' A2 Ae
0r Sr r dr
dCD
dr
d 2
Ca
"
A
2 Ae
2 AnaC A
2
Ana
'
A
2
An
"
A
CkSr
CkSr
Ckr
0r SCk 2Ca
n An
rate equationdefinitions
substitute Fick
s Law
Step 3: Write rate laws
Step 2: Write the expression for W Ar
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Diffusion/Rxn in Porous Catalysts
0r SCkr dr
dCDdr d 2
Can An
2 Ae
finiteC 0r A symmetry
AsRr A CC surface
Step 5a: Write the equationin dimensionless form
As
A
CC
Rr
As A C
ddC
R1
ddr
RC
dd
dr dC As A
0CDSk
dr dC
r 2
dr Cd n
Ae
Can A2 A
2
0D
CRSkdd2
dd n
e
1n As
2Can
2
2
Step 4: Identify boundary conditions
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Diffusion/Rxn in Porous Catalysts
Step 5b : define Thiele modulus (n)
0D
CRSkdd2
dd n
e
1n As
2Can
2
2
e
1n
As
2
Can2n D
CRSk
0dd2
dd n2
n2
2
understand the Thiele modulus
R0CDRCSk
Ase
n AsCan2
nr eac t ion r a t e
d i f f u s i o n r a t e
large n - diffusion controls
small n - kinetics control
0CD
Sk
dr
dC
r
2
dr
Cd n A
e
Can A
2
A2
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Diffusion/Rxn in Porous Catalystsfirst orderkinetics(n = 1)
define y =
0dd2
dd 2
12
22
e
Can21 R
DSk
322
2
2
2 y2ddy2
dyd1
dd
2
y
d
dy1
d
d
0yd
yd 212
2
1111 sinhBcosh Ay
1B
1 A sinhcosh 11
differential has the solution apply boundary conditions
1 ,1
finiteis ,0
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Diffusion/Rxn in Porous Catalystsfirst orderkinetics(n = 1)
0dd2
dd 2
12
2 2
e
Can21 R
DSk
0yd
yd 212
2
1111 sinhBcosh Ay
1B
1 A sinhcosh 11
differential has the solution apply boundary conditions
1 ,1
finiteis ,0
1
1
sinhsinh1
As
A
CC
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Thiele Modulus
As
A
CC
Refer Figure 12-4 pg 823for concentrationprofile
d 1 1 sinh 1 sinh 1
00.510
0.5
1
1
2 5 20
Go thru’ Example 12 -2 pg823
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The internal effectiveness factor ( ) is a measure of therelative importance of diffusion to reaction limitations :
Internal Effectiveness Factor ( )
s As T,Ctoexposedweresurfaceentireif rateratereactionoverallactual
As
A"
As
" A
' As
' A
As
A
MM
r r
r r
r r M mol / time
r mol / time / mass cat
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Internal Effectiveness Factor ( )
Determine M As (rate if all surface at C As)
catalystmass
catalystmass
areasurfaceareaunitper rateM As
' Asr
aS
CV AsM
x
x
As1Ck
c3
34
a As1 As RSCkM
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Determine M A (actual rate is equal to reactant diffusionrate at outer surface)
Internal Effectiveness Factor ( )
1 Ase A d
dCRD4M
11
12
1
11
1 sinhsinh1
sinhcosh
dd
1coth 11
1cothCRD4M 11 Ase A
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Substitute results into definition of
Internal Effectiveness Factor ( )
As
A
MM
c
3
3
4
a As1
11 Ase
RSCk1cothCRD4
1cothRSk
D3 11
c2
a1
e
1coth3112
1
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Internal Effectiveness Factor ( )
1coth3
1121
0.1 1 10 1000.1
1
12
131
21
101
ac1
e21
SkD
R33
1 20
small d p
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Internal Effectiveness Factor ( )
1coth3
1121
0.1 1 10 1000.1
1
12
131
21
101
ac1
e21
SkD
R33
1 20
reactionrate
limited
internaldiffusionlimited
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22.21
Effectiveness Factor
Note that internal diffusion resistance decreases with decreasing. Therefore, the influence of diffusion on the reaction rate
supported by a particle is reduced when particle radius is reduced,DAB is high and the rate constant is relatively small.
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Thiele modulus - Derived for spherical particle geometry
Derived for 1 st order kineticsFor large , approximately
Internal effectiveness factor - Assumed =0, correction applied when 0Assumed isothermal conditions
Revisit and
21
213
1n2
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For exothermic reactions, can be > 1 as internaltemperature can exceed T s.The rate internally is thus larger than at the surface
conditions where is evaluated.The magnitude of this effect is dependent on
Hrxn , Ts, Tmax , and k t (thermal conductivity of the pellet)
and are used to quantify this effect:
can result in mulitple steady statesNo multiple steady states exist if Luss criterion is fulfilled
Non-Isothermal Behavior
Number ArrheniussRT
Est
Aserxn
s
smax
TkCDH
TTT
14
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When both internal AND external diffusion resistances areimportant (i.e., the same order of magnitude), both must be
accounted for when quantifying kinetics.It is desired to express the kinetics in terms of the bulkconditions, rather than surface conditions:
Overall Effectiveness Factor
bulk A,Ctoexposedweresurfaceentireif rateratereactionoverallactual
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Accounting for reaction both on and within the pellet, themolar rate becomes:
For most catalyst, internal surface area is significantly higherthan the external surface area:
Overall Effectiveness Factor
V1Sar M
cac
"
A A
b
bac"
Ac A Sar aW
ba"
Ac A Sr aW
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Overall Effectiveness Factor
ba"
Ac A Sr aW reaction rate(internal & external surfaces)
VaCCkVaW c Asbulk, Acc Ar mass transport rate
internal surfaces notall exposed to C As
As1"
As"
A Ckr r Relation between C As and C A defined by the as:
VSCkVaW ba As1c A
ba As1c Asbulk, Ac SCkaCCk
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Overall Effectiveness Factor
ba"
Ac A Sr aW reaction rate(internal & external surfaces)
VaCCkVaW c Asbulk, Acc Ar mass transport rate
As1"
As"
A Ckr r Relation between C As and C A defined by the as:
ba1cc
bulk, Acc As Skka
CkaCSolving for C As :
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Overall Effectiveness Factor
ba"
Ac A Sr aW reaction rate(internal & external surfaces)
VaCCkVaW c Asbulk, Acc Ar mass transport rate
ba1cc
bulk, Acc1" A
Skka
Cakkr Substitution into the rate law:
ba1cc
bulk, Acc As Skka
CkaCSolving for C As :
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Overall Effectiveness Factor
summary of factor relationships:
ba1cc
bulk, Acc1" A Skka
Cakkr
Rearranging the expression:
bulk, A1ccba1
CkakSk1
"bulk, A
" A r r ccba1 akSk1
" As
"bulk, A
" A r r r
As1"
As Ckr
Ab1"
Ab Ckr
Overall Effectiveness Factor ( )
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Weisz-Prater Criterion is a method of determining if a givenprocess is operating in a diffusion- or reaction-limited regime
CWP is the known as the Weisz-Prater parameter. All quantities are
known or measured.
CWP << 1, no C in the pellet (kinetically limited)CWP >> 1, severe diffusion limitations
Weisz-Prater Criterion
Ase
c2'
obs, A21WP CD
Rr C
Go thru’ Example 12 -3 pg 839
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Mass transfer effects negligible when it is true that
n is the reaction order, and the transfer coefficients k c and h(below) can be estimated from an appropriate correlation(i.e., Thoenes-Kramers for packed bed flow)
Heat transfer effects negligible when it is true that
Mears’
Criterion
15.0
Ck
nRr
Abc
b'
A
15.0ThR
REr H2
bg
b'
Arxn
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0r dz
dCU
dzCd
D A'
A Ab
2 Ab
2
AB
Shell balance onvolume element A z
Mole flux of A
First order reaction
Application to PBRs – Mass Transferand Reaction
0r dz
dWb
' A
Az
UCdz
dCDW Ab A
AB Az
Aba'
Ab'
A CkSr r
0CkS Abab
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0CkSdz
dCU
dzCd
D Abab Ab
2 Ab
2
AB
Axial dispersion negligible(relative to forced axialconvection) when…
dp is the particle diameterUo is the superficial velocity of the gasDa is the effective axial dispersion coefficient
Application to PBRs
a
po
Abo
pb'
A
D
dU
CU
dr
Which can be rewritten as:
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Application to PBRs
Which can be rewritten as:
Abab Ab C
UkS
dzdC
Entrance condition:o Ab0z Ab CC
Integrating and applying boundary condtion yields:
UzkS
expCC ab Ab Ab o
Go thru’ Example 12 -4 pg 845
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Problem in Class (pg 860-Fogler)