10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 17: Mass Transfer Resistances
This lecture covers: External diffusion effects, non-porous packed beds and
monoliths, and immobilized cells
Table 1. Homogeneous vs. Heterogeneous Catalysis
Homogeneous
vs.
Heterogeneous Catalysis
acids,bases
immobilized enzymes
radicals
metals
organometallics
solid acids, bases
enzymes
metal oxides, zeolites, clays, silica
better mixing, uniformity
multiphase systems
transport limitations
reuse catalyst easily
product purity
1. New rate law on surface
2. Model the transport and mixing
Ci fluid
( x, y,z ) =
θ jsurface
( x, y ) =
Ni
Volume
N j on surface
N sites on surface
∑ θ j + θvacancy = 1
ri
fluid
=
N rxn fluid
∑
n =1
⎛ A⎞
ν i ,n rn (C ) + ⎜ ⎟
⎝V ⎠
N rxn suface
∑
m =1
ν i ,m rm" (C ,θ )
mol
s ⋅ vol
mol
s ⋅ area
QSSA for surface species: rj =
"
∑ν
r (C ,θ ) ≈ 0
"
j ,m m
θQSSA = f (C )
At surface
dθ j
0
⎛ area ⎞
= ( flows ) + rj" ⎜
⎟ N avagadro
dt
⎝ N sites ⎠
A
A
P
δ
surface
Catalyst
Figure 1. Schematic of boundary layer at catalyst surface for a
turbulent, well mixed system where CA is a function of x.
Cite as: William Green, Jr., 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].
(
Flux of A to the surface: WA = −Ctot D∇ y A + y A WA + WP
)
where yA is the mole fraction of A
Ctot = const
diffusion
convection
P
WA = − D∇C A + C A u
FAnet = − v∫ WA ⋅ dn
Åsurface integral
JK
FA = ∫∫∫ dxdydz ∇ ⋅ WA
JK
dN A
= ∫∫∫ dxdydz ±∇WA + rA ( x, y, z )
dt
dC A JK
= ∇ ⋅ WA + rA
dt
(
)
*See Fogler 11-21
Continuity equation:
dC A
= D∇ 2C A − u ⋅∇C A + rA (C )
dt
Boundary Condition:
WAinto wall = −rA'' at surface
1. Steady state
2. Gradients
dC A
dC A
and
are negligible
dx
dy
3. Velocity u towards the wall = 0
4. No reaction in the fluid
∂ 2C A
∂z 2
WA = −rA''
0=D
−D
dC A
dz
= − rA'' ( C z =0 ) (z=0 at the surface)
z =0
⎡ C main − C A ( z = 0) ⎤
C A ( z ) = C A ( z = 0) + ⎢ A
z⎥
δ
⎣
⎦
main
dC
C
− C A ( z = 0)
=D A
= −rA'' ( C A, z =0 )
D A
dz z =0
δ
Slow chemistry limit: C A ( z
= 0) ≈ C Amain
(
rA'' ≈ rA'' C Amain
Fast chemistry limit: C A ( z
D
δ
)
= 0) ≈ 0
C Amain ≈ rA''
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 17
Page 2 of 3
Cite as: William Green, Jr., 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].
D
δ
= kc “mass transfer coefficient”
Sh =
kc d p
D
ÅSherwood number (dimensionless)
For spherical, catalyst particle with diameter dp:
Sh = 2 + 0.6 Re1 2 Sc1 3
ud
υ
μ
Sc =
Re = p υ =
D
υ
ρ
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 17
Page 3 of 3
Cite as: William Green, Jr., 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].