1. L 29-Heterogeneous Catalysis and Reactor Design
Prof. K.K.Pant
Department of Chemical Engineering
IIT Delhi.

2. Diffusion and Reaction in a spherical pellet (Reading assignment: Fogler, Ch 12))
In – out – disappearance =0 (spherical shell balance)
WAr (4 πr2)r - WAr (4 πr2)r+∆ r - rA’ (4 πr2 c ∆r) =0
Dividing by -4 π ∆ r
=r+Δr
Moles = WAr (4 πr2)r
Molar flux
r=0, CA finite, r=R, CA=CAS
-rA=c(-r’A)
Boundary conditions

3. consider 1st order What about n-th order ?  c(-r’A) =-rA -rA=kCA
Differentiation &
Divide by –r2De
What about n-th order ? 
Differentiation &
Divide by –r2De

4. Dimensionless eq. – 1st order
Thiele Module
About for n-th order ?
Thiele Module

5. Y =  , , =>= y/ 
d/ d = 1/ (dy/d)- y/ 2
d2/ d2 = 1/ (d2y/d2)- 2/ 2 dy/d + 2y/3
d2y/d 2 - ø2y=0
y= A Cosh ø  + B Sinh ø 
A=0 as φ must be finite at the centre,
(B. C. =0, coshø 1; and
Sinh ø 0.
And at =1, =1,=> B= 1/Sinh ø
Thus , = CA/CAs = 1/  [Sinh ø  / Sinh ø]

6. Thiele Modulus, n If n is large – internal diffusion limits the
overall rate
If n is small – the surface reaction limits the

7. Internal Effectiveness Factor
Internal effectiveness Factor,  is: ranged 0 – 1
 for a first-order reaction in a spherical catalyst pellet

8. Calculation of Catalytic Effectiveness Factor
η = Actual overall rate(RA /Rate in the absence of diffusion resistance (RAs )
Global rate RA = 4 πR2 De (dCA/dr) at r=R
Or RA = 4 πR De (d  /d ) at  =1
((d  /d  )at  =1 = (ø cot h ø-1)
RA = 4 πR De CAS (ø cot h ø-1) Global Rate.
Thus η = [4 πR De CAS (ø cot h ø-1)] / k’ ρc CAS 4/3 πR 3
η = 3 (ø cot h ø-1)/ k’ ρc R2/De R
η = 3 (ø cot h ø-1)/ ø2
for ø> 20, η= 3/ ø strong pore diffusion resistance

9. Calculation of Catalytic Effectiveness Factor
where
- Thiele Modulus
1st order reaction rate:
Spherical Pellet
Cylindrical Pellet
Slab Pellet

10. Internal Effectiveness Factor

11. Weisz – Prater Criterion for internal diffusion
Uses the measured values of the rate of reaction to determine if Internal diffusion controls the rate.
Weisz-Prater Parameter CWP
ηø2 = 3(Ø Coth Ø-1)
ηø2 = (observed rate/rate cal. at CAS) x (rate calculated at CAS) / diffusion Rate)
η = (-r’A(obs)/ -r’As
Ø2= -r”AS Sa ρp R2/De CAs = -r’AS ρp R2/De CAs

12. CWP= (-r’A(obs)/ -r’As )(-r’AS ρp R2/De CAS)
CWP = (-r’A(obs) (ρp R2/De CAS)
These are measured or known terms.
if CWP << 1, No diffusion limitations and no concentration gradient exists in the pellet.
CWP >> 1, Internal diffusion limits the rate.

13. Non isothermal pellet Energy balance
Mass Balance

14. Non isothermal pellet effectiveness factor

15. Falsified Kinetics
Measurement of the apparent reaction order and activation energy results primarily when internal diffusion limitations are present.
This becomes serious if the catalyst pellet shape and size between lab (apparent) and real reactor (true) regime were Too different.
Smaller catalyst pellet  reduces the diffusion limitation  higher activation energy  more temperature sensitive 
RUNAWAY REACTION CONDITIONS!!!!

16. Falsified Kinetics
With the same rate of production, reaction order and activation energy to be measured (apparent rate)

17. Overall effectiveness factor (Both internal and external diffusion are important

18. Overall Effectiveness Factor
At Steady state, Moles transported from bulk fluid to the external surface of the Catalyst (WAr Ap )= Net Rate of reaction with in and on the pellet,
MA = WAr Ap = -r”A(As+ Ap),=( Molar Flux x Ext. Surface Area of pellet)
For a single spherical pellet of Radius R,
AP= 4π R2, and As= SA x mass of pellet,(As >>Ap)
AP = (ext. SA/reactor volume) (reactor volume) = ac ∆V
As= (int. SA/.mass of catalyst) (mass cat./vol. cat) (vol cat/reactor vol.) . Rect vol.) => AS= SA ρ c (1- ø) ∆V