Institut für Chemie
Mass Transfer in Heterogeneous Catalysis
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Reinhard Schomäcker
Institut für Chemie
der Technischen Universität Berlin
For more details see:
Roland Dittmeyer and Gerhard Emig
Simultaneous Heat and Mass Transfer and Chemical Reaction,
chapter 6.3 in Handbook of Heterogeneous Catalysis Eds. G. Ertl, et.al
Wiley-VCH, Weinhein, 2007
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Molecular Level: e.g. Langmuir-Hinshelwood Kinetics
A
B
K A pA
θA =
1 + K A pA
Adsorption
Reaction
Desorption
= sequential steps
pA= partial pressure near surface
k K A p A K B pB
r = k θ A θB =
(1 + K A p A + K B pB )2
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Institut für Chemie
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Why so different
geometries?
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τ = VR/VF
τ = VR/q uF
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Pressure drop in a fixed bed
channel
1 d 2 Δp
u=
32 η L
2
k
d Δp u0
u = F (ε )
=
η L ε
1
ε 3 d k2 Δp
u0 =
150 (1 − ε )2 ηL
2
(
1− ε )
λ (Re ) =
ε3
laminar
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300
Re
1 ⎛ ε ⎞
F (ε ) =
⎜
⎟
150 ⎝ 1 − ε ⎠
2
L
1 2
Δp = λ (Re ) ρu0
dk
2
(
1 − ε ) ⎡ 300(1 − ε )
⎤
λ (Re ) =
+ 3,5
ε3
⎢⎣
Re
⎥⎦
laminar und turbulent, Ergun-Eq.
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Sequential steps:
Pressure driven flow,
Film diffusion,
Pore diffusion
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Steps of Heterogeneous Reaction
1. 
2. 
3. 
4. 
5. 
6. 
7. 
Diffusion of reactant to catalyst
Transport of reactant within catalyst pores
Adsorption of reactant on catalyst surface
Reaction
Desorption of products from catalyst surface
Transport of products out of catalyst pores
Diffusion of products away from catalyst
Transport and reaction occure simultaneously
(at a catalyst under steady state conditions)
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Impact of mass transfer limitation
Feed
k>D
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Feed
k<D
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Mass Transport and Heterogeneous Catalysis
Surface layer
Principles
fluid phase
catalyst
Concentration profile
Mass transport
influence
Influence of mass transport on the temperature dependance of het. catalysis
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Institut für Chemie
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Time Scales in a Reactor
residence time τ = VR/VF
Mixing time
Θ
Time constant of reaction = CM,0/R0
Time constant of diffusion = R2/De
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Description of pore diffusion
grad c
→
j = − D grad c
→
1. Fick`s Law
Dg =
j
A
1
Λw
2
Λ=
k BT
2 πσ 2 p
Average free path length
w=
8 k BT
πm
Average molecular velocity
Dg ~ T1,5
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und Dg ~ 1/p
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a) Diffusion in pores dp >> λ
D eff = D g
ε
τ
b) Knudsen – Diffusion
D eff
Tortuosity
τ≈
L
X
dp = λ
ε 1 dp ε
= DK =
w
τ 2 2 τ
DK ~ T0.5
c) Intermediate range
D eff =
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1
ε
1
1 τ
+
Dg DK
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a) Diffusion in pores dp >> λ
D eff = D g
ε
τ
b) Knudsen – Diffusion
D eff
Wicke-Kallenbach-Experiment
N 2, X
N2
dp = λ
porous
material
ε 1 dp ε
= DK =
w
τ 2 2 τ
DK ~ T0.5
c) Intermediate range
D eff =
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N2, X =?
1
ε
1
1 τ
+
Dg DK
→
j
x
= − Degradc x
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Simultaneous reaction and pore diffusion in a sperical particle
temporal change
of materials within
volume element
=
changes of material
amount by transport
+
changes caused
by reactions
Mass balance
dV
 
dci
∫ dt dV = − ∫ ( ji do) + ∫ νi r dV
V
V
 

∫ ( j do) = ∫ div j dV
V
→
dc i
= − div ji + νi r
dt
one dimensional
Spherical geometry
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→
div j =
→
div j =
d jz
dz
1 d
2
(
R
jR )
2
R dR
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0=−
1 d
2
(
R
ji R ) + νi r
2
R dR
Mass balance of sperical particle
in steady state
Solution of mass balance and description of average reaction rate
d 2 c 2 dc
Deff ( 2 +
) = k cn
R dR
dR
with r= kcn and ν=-1
c( R = R 0 ) = c 0
dc
R =0 = 0
dR
kc 0n−1
Deff
renormalized parameter
= Thiele-Modulus
R
Φ0 )
R0
sinh Φ0
radial concentration profil
within sperical pellet
Φ0 = R 0
R
c( R ) = c0 0
R
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sinh(
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k<D
k>D
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rVs = As Deff
rhet
dc
dR
r = rhet
R =R0
3
co
Φ0
=
D eff
(
− 1)
R0
R 0 tanh Φ 0
rhom = k c 0
η=
Porenwirkungsgrad
effectiveness
factor
1
Reaction without mass transport limitation
rhet
3
1
1
=
(
−
)
rhom Φ 0 tanh Φ 0 Φ 0
Effectiveness factor
0,8
0,6
0,4
0,2
0
0
5
10
15
20
Thiele-Modul
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Influence of temperature
3
η≈
Φ0
rhet
3
3
=
k c0 =
Φ0
R0
k eff ≅ Deff k
Deff
k c0 = k eff c0
k
E eff =
1
(E + ED )
2
Influence of pore diffusion on effective
rate constant
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Film diffusion und Reaction
nDiff = Sa D c0 − c W
δ
.
n Diff = Sa β( c 0 − c W ),
⎡ mol ⎤
⎢
3 ⎥
⎢⎣ s * m ⎥⎦
1 1 dn i
r=
νi V dt
r = rSa ,
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k = k Sa
1 1 dn i
rS =
νi Sa dt
D
mit β =
δ
⎡ mol ⎤
⎢
2 ⎥
⎢⎣ s * m ⎥⎦
Sa
mit: a =
V
Block Course “Reactivity and Catalysis”
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.
n = Sa β (c0 − cW ) = Sa rS (cW )
Discussion of a first order reaction with rs=kscw
ß(c0-cw) = kscw
c 0β
cW =
β + kS
Border cases:
ks << ß
cw =c0
ks >> ß
cw ≈ 0
c0
( no layer formation)
cw
c0
cw
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βk S
rS = k S cW =
c0 = k S ,eff c0
β + kS
1
k S,eff
1 1
=
+
kS β
Calculation of eff. volume related rate constant
k eff = a k S,eff =
a =
1 1
+
ks β
Particle surface
Particle volume
1
1
1
=
+
k eff aβ ηk
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a
=
1
1 1
+
k aβ
π
=
π
6
d
d
2
3
=
6
dsphere
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Temperature dependance:
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Criteria to exclude mass transfer limitations
Generalized Thiele-Modulus
Criterion with mesasurable data only: Weisz-Modulus
<1
1
effectiveness
Porenwirkungsgrad
factor
0,8
0,6
0,4
0,2
0
0
5
10
Thiele-Modul
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Experimental check
for pore diffusion
Reduce particle size
decrease φ
increase η το 1
for film diffusion
Decrease reactor diameter
τ=VR/Vf
increase u = Vf/q
increase ß = D/δ
at constant τ, Τ, ci
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at constant τ, Τ, ci
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