L 29-Heterogeneous Catalysis and Reactor Design

Prof. K.K.PantDepartment of Chemical Engineering

IIT Delhi.kkpant@chemical.iitd.ac.in

Diffusion and Reaction in a spherical pellet (Reading assignment: Fogler, Ch 12))

dr

dCD

dr

dycDW A

eA

eAr

0)( 22

rrdr

rWdcA

Ar

0])/([ 22

rrdr

rdrdCDdcA

Ae =r+Δr

Moles = WAr (4 πr2)r

Boundary conditions

-rA=c(-r’A)

r=0, CA finite, r=R, CA=CAS

Molar flux

In – out – disappearance =0 (spherical shell balance)

WAr (4 πr2)r - WAr (4 πr2)r+∆ r - rA’ (4 πr2 c

∆r) =0Dividing by -4 π ∆ r

consider 1st order

0])/([ 22

rrdr

rdrdCDdcA

Ae

0])/([

12

2

AAe Ckr

dr

rdrdCDd

c(-r’A) =-rA

-rA=kCA

What about n-th order ?

0])/([ 22

n

AnAe Ckr

dr

rdrdCDd

Differentiation &Divide by –r2De

Differentiation &Divide by –r2De

Dimensionless eq. – 1st order

About for n-th order ?

02

2

2

n

Ae

nAA CD

k

dr

dC

rdr

Cd

02 1

2

2

A

e

AA CD

k

dr

dC

rdr

Cd 2212

20

d d

d d

22

2

20n

n

d d

d d

Thiele Module

Thiele Module

eD

Rk 21

e

nAsn

D

CRk 12

5

Y = , , =>= y/ d/ d = 1/ (dy/d)- y/ 2

d2/ d2 = 1/ (d2y/d2)- 2/ 2 dy/d + 2y/3 Þd2y/d 2 - ø2y=0Þy= A Cosh ø + B Sinh ø ÞA=0 as φ must be finite at the centre, (B. C. =0, coshø 1; and ÞSinh ø 0.And at =1, =1,=> B= 1/Sinh ø

Thus , = CA/CAs = 1/ [Sinh ø / Sinh ø]

2212

20

d d

d d

Thiele Modulus, n

2 n-1 n2 n As n Asn

e e As

k R C k RC "a" surface reaction rateφ = = =

D D [(C -0)/R] "a" diffusion rate

A 1

As 1

C sinhφ λ1ψ = =

C λ sinhφ

• If n is large – internal diffusion limits the overall rate• If n is small – the surface reaction limits the overall rate

Internal Effectiveness Factor

• Internal effectiveness Factor, is: ranged 0 – 1

• for a first-order reaction in a spherical catalyst pellet

As s

Actual overall rate of reactionη =

Rate of reaction that would result if entire

interior surface were exposed to the external

pellet surface conditions C ,T

' "A A A

' "As As As

-r -r -rη = = =

-r -r -r

1 121

3η = φ cothφ -1

φ

Calculation of Catalytic Effectiveness Factor

)1(3 Coth

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

Þ ((d /d )at =1 = (ø cot h ø-1)

ÞRA = 4 πR De CAS (ø cot h ø-1) Global Rate.ÞThus η = [4 πR De CAS (ø cot h ø-1)] / k’ ρc CAS 4/3 πR 3

Þ η = 3 (ø cot h ø-1)/ k’ ρc R2/De RÞ η = 3 (ø cot h ø-1)/ ø2

for ø> 20, η= 3/ ø strong pore diffusion resistance

Calculation of Catalytic Effectiveness Factor

Catalytic Effectiveness Factor:

wheref- Thiele Modulus

1st order reaction rate:

Spherical Pellet

Cylindrical Pellet

Slab Pellet

)313(1 Coth

DekSaRp /

3

DekSaRp /

2

DekSaL p /

Internal Effectiveness Factor

11

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

Weisz – Prater Criterion for internal diffusion

12

ÞCWP= (-r’A(obs)/ -r’As )(-r’AS ρp R2/De CAS)

Þ CWP = (-r’A(obs) (ρp R2/De CAS)

Þ These are measured or known terms.

Þ if CWP << 1, No diffusion limitations and no concentration gradient exists in the pellet.

ÞCWP >> 1, Internal diffusion limits the rate.

13

Non isothermal pellet Energy balance

0])/([ 22

rrdr

rdrdCDdcA

Ae Mass Balance

Energy balance

14

Non isothermal pellet effectiveness factor

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!!!!

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