Reinhard Schomäcker Institut für Chemie der Technischen

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Mass Transfer in Heterogeneous Catalysis

Reinhard SchomäckerInstitut für Chemieder Technischen Universität Berlin

For more details see: Roland Dittmeyer and Gerhard EmigSimultaneous Heat and Mass Transfer and Chemical Reaction, chapter 6.3 in Handbook of Heterogeneous Catalysis Eds. G. Ertl, et.alWiley-VCH, Weinhein, 2007



Molecular Level: e.g. Langmuir-Hinshelwood Kinetics

A B

AA

AAA pK

pK

+=

1θ

( )21 BBAA

BBAABA

pKpK

pKpKkkr

++== θθ

AdsorptionReactionDesorption

= sequential steps

pA= partial pressure near surface











Pressure drop in a fixed bed

ud p

L= 1

32

2

η∆

2

1150

1)(

−=

εεεF

( ) L

pdu k

ηεε ∆−

=2

2

3

01150

1 ( ) 202

1Re u

d

Lp

k

ρλ=∆

( ) ( )Re

Re3001

3

2

εελ −= ( ) ( )

+−−= 5,3)1(3001

3 ReRe

εε

ελ

laminar laminar und turbulent, Ergun-Gl.

( )εη

ε 02 u

L

pdFu k =∆=



Sequential steps:

Pressure driven flow,Film diffusion,Pore diffusion



Steps of Heterogeneous Reaction

1. Diffusion of reactant to catalyst

2. Transport of reactant within catalyst pores

3. Adsorption of reactant on catalyst surface

4. Reaction

5. Desorption of products from catalyst surface

6. Transport of products out of catalyst pores

7. Diffusion of products away from catalyst

Transport and reaction occure simultaneously(at a catalyst under steady state conditions)



Impact of mass transfer limitation

k > D k < D

Feed Feed



Mass Transport and Heterogeneous Catalysis

PrinciplesSurface layer

catalyst

Concentration profile

fluid phase

Influence of mass transport on the temperature dependance of het. catalysis

Mass transportinfluence







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



Description of pore diffusion

cgradDj −=→

A

→j

cgrad

D wg = 1

2Λ

Λ =

=

k T

p

wk T

m

B

B

2

8

2πσ

π

Average free path length

Average molecular velocity

Dg ~ T1,5 und Dg ~ 1/p

1. Fick`s Law



a) Diffusion in pores dp >> λ

D Deff g= ετ

b) Knudsen – Diffusion dp = λ

D Dd

weff Kp= =ε

τετ

1

2 2

DK ~ T0.5

c) Intermediate range

D

D D

eff

g K

=+

11 1

ετ

N2, X

N2, X =?

N2

Wicke-Kallenbach-Experiment

porousmaterial

cDj xexgrad−=

→



Simultaneous reaction and pore diffusion in a sperical particle

temporal change changes of material changes causedof materials within = amount by transport + by reactionsvolume element

dV

dcdt

dV j do r dVi

Vi i

V∫ ∫ ∫= − +( )

r rν ( )∫ ∫=

r r rj do div j dV

V

dc

dtdiv j ri

i i= − +→

ν

one dimensional

Spherical geometry

div jd j

dzz

→=

div jR

d

dRR jR

→= 1

22( )

Mass balance



01

22= − +

R

d

dRR j ri R i( ) ν Mass balance of sperical particle

in steady state

Solution of mass balance and description of average reaction rate

Dd c

dR R

dc

dRk ceff

n( )2

2

2+ = with r= kcn and ν=-1

c R R c

dc

dR R

( )= =

==

0 0

0 0

Φ0 00

1

=−

Rkc

D

n

eff

renormalized parameter= Thiele-Modulus

c R cR

R

RR

( )sinh( )

sinh= 0

0 00

0

Φ

Φ

radial concentration profilwithin sperical pellet



k > D

k < D



0RReffss dR

dcDAVr

=

= r = rhet

rR

Dc

Rhet effo= −3

10 0

0

0

(tanh

)Φ

Φ

r k chom = 0 Reaction without mass transport limitation

η = = −r

rhet

hom

3 1 1

0 0 0Φ Φ Φ(tanh

) Effectiveness factor

Thiele-Modul

Po

ren

wir

kun

gsg

rad

0

0,2

0,4

0,6

0,8

1

0 5 10 15 20

effe

ctiv

enes

sfa

cto

r





η ≈ 3

0Φ

r k cR

D

kk c k chet

effeff= = =3 3

00

00 0Φ

k D keff eff≅ E E Eeff D= +1

2( )

Influence of pore diffusion on effectiverate constant

Influence of temperature



Film diffusion und Reaction

ms

mol3

*

&n S Dc c

Diff aW= −0

δ

n S c c mitD

Diff a W

.( ),= − =β β

δ0

rV

dn

dti

i= 1 1

νr

S

dn

dtSi a

i= 1 1

ν

ms

mol2

*

r r a k k a mit aS

VS Sa= = =, :



)()( 0

.

WSaWa crSccSn =−= β

Discussion of a first order reaction with rs=kscw

ß(c0-cw) = kscw

cc

kWS

=+0β

β

Border cases:

ks << ß cw =c0 ( no layer formation)c0 cw

ks >> ß cw 0 c0

cw

≈



0,0 ckck

kckr effS

S

SWSS =

+==

ββ

1 1 1

k kS eff S,

= +β

k a ka

k k a

eff S eff

s

= =+

=+

, 1 11

1 1β β

Calculation of eff. volume related rate constant

1 1 1

k a keff

= +β η

dspheredd

Particle volumea

6

63

2

=== ππParticle surface



Temperature dependance:



< 1

Generalized Thiele-Modulus

Criterion with mesasurable data only: Weisz-Modul

Criteria to exclude mass transfer limitations

T h i e le - M o d u l

Po

ren

wir

kun

gsg

rad

0

0 , 2

0 , 4

0 , 6

0 , 8

1

0 5 1 0 1 5

effe

ctiv

enes

sfa

cto

r



Experimental check

for pore diffusion for film diffusion

Reduce particle sizedecreaseφincreaseη το 1

Decrease reactor diameter

at constantτ, Τ, ci

τ=VR/Vf

increase u = Vf/q

at constantτ, Τ, ci

increase ß = D/δ





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