10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 20: Reaction and Diffusion in Porous Catalyst
This lecture covers: Effective diffusivity, internal and overall effectiveness factor,
Thiele modulus, and apparent reaction rates
Reaction & Diffusion
-Diffusion in a porous solid phase
Ex. Precious metals on ceramic supports or drug/nutrient delivery through tissues
-Derive steady state material balance accounting for diffusion and reaction in a
spherical geometry
-Thiele modulus (φ)
r
R
[S]0= surface concentration of a growth
substrate (ex. glucose and O2)
Figure 1. Sphere of Cells
Assume pseudo-homogeneous medium and Fick’s Law describes diffusion
Flux = − D
d [S ]
dr
, where
Flux [=]
# molecules
length 2
and D [=]
time
area ⋅ time
− rs = knCsn
V C
− rs = max s
K s + Cs
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
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-rs
n=1
n=0
Cs
Figure 2. Rate of reaction versus species concentration
Cs K s ⇒ 1st order rs ≈
Vmax Cs
Ks
Cs K s ⇒ 0th order rs ≈ Vmax
Steady-state Shell Balance
In some cases, S still hasn’t
penetrated to the center
t=0 Æ
tÆ ∞
Figure 3. Time progression as species, S, enters the sphere
Thin shell r to (r+Δr)
Sin by diffusion –Sout by diffusion –Scons by reaction=0
Flux ⋅ 4π r 2
r + Δr
− Flux ⋅ 4π r 2 − kn Csn 4π r 2 Δr = 0
r
Divide through by 4πΔr and take the limit as ΔrÆ0
(
d Flux ⋅ r 2
dr
) −k C r
n
n 2
s
=0
dC
d ⎛
⎞
− D s ⋅ r 2 ⎟ − knCsn r 2 = 0
⎜
dr ⎝
dr
⎠
⎫
d 2Cs 2 dCs kn n
+
− Cs = 0 ⎬ 2nd order ODE
2
dr
r dr D
⎭
2 Boundary conditions:
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 20
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Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
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Cs
r=R
= Cs ,0
Cs is finite everywhere, or
dCs
dr
=0
r =0
Nondimensionalize
C
r
S= s
R
Cs ,0
ρ=
d 2 S 2 dS
+
− φ 2S n = 0
dρ2 ρ dρ
Boundary conditions: S=1 @ ρ=1
S is finite everywhere
φ =
2
kn R 2Csn,0−1
D
=
(R
2
D
)
(1 k C )
n
n −1
s ,0
=
characteristic diffusion time
characteristic reaction time
If diffusion is slow Æ diffusion dominates
If reaction is slow Æ reaction dominates
If φ2<<1 Æ reaction limited regime
If φ2>>1 Æ diffusion limited regime
φ2<<1
substrate
throughout
φ2>>1
substrate can’t make
it to the center
Figure 4. Reaction and diffusion limited regimes
S=
1 sinh(φρ )
ρ sinh(φ )
sinh( z ) =
e z − e− z
2
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 20
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2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
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1
φ = 0.1
S
φ =1
φ↑
φ = 10
ρ
1
Figure 5. S versus ρ for various values of φ
Define “effectiveness factor” η
overall rate of reaction
η=
rate if Cs =CS,0 everywhere
overall reaction rate in sphere at steady state = [inward flux @ r=R (ρ=1)]*Area
=D
dCs
dr
4π R 2
r=R
= 4π RDCs ,0
η=
3
φ2
dS
dρ
= 4π RDCs ,0 (φ coth φ − 1)
ρ =1
(φ coth φ − 1)
1
η
0.1
η∝
1
φ
0.01
.01
0.1
1
10
100
φ
Figure 6. Log-log plot of effectiveness factor versus thiele modulus
Higher values of Thiele modulus Æ effectiveness goes down
*For a variety of reaction kinetics, geometries and rate laws, plots of η vs φ all look
the same.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 20
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Shrinking Core Model
In cases with noncatalytic and irreversible reaction, diffusion limit is describable by
the “shrinking core model”.
Figure 7. Shrinking core model
Rapid, irreversible reaction limited by rate of diffusion of a reactant from the surface
The following must be written down for the shell balance:
1. Rate of reaction
2. Rate of diffusion
3. Rate of movement of the core
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 20
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Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].