8/13/2019 CRE II -27

1/17

L -27 Diffusion Effects in Spherical Pellet

Prof. K.K.Pant

Department of Chemical EngineeringIIT Delhi.

kkpant@chemical.iitd.ac.in

mailto:kkpant@chemical.iitd.ac.inmailto:kkpant@chemical.iitd.ac.in

8/13/2019 CRE II -27

2/17

8/13/2019 CRE II -27

3/17

3

Solving

The concentration profile in a pore

How much reaction rate is lowered because of resistance to pore diffusion

8/13/2019 CRE II -27

4/17

4

Distribution and average value of reactant concentration within a catalyst

pore as a function of the parameter Thiele modulus

m L= =

8/13/2019 CRE II -27

5/17

5

Diffusion and Reaction in a Spherical Pellet

Rate

8/13/2019 CRE II -27

6/17

6

8/13/2019 CRE II -27

7/17

Derivation of the Differential Equation

~ Diffusion and Reaction in a spherical pellet

dr

dCD

dr

dycDW Ae

AeAr

0)( 2

2

rrdr

rWdcA

Ar

0])/([ 22

rrdr

rdrdCDdcA

Ae

=r+r

Moles = WAr(4 r2)r

Boundary

conditions

Order of

reaction?-rA=c(-rA)

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

Molar flux

Inoutdisappearance =0

WAr(4 r2)r - WAr(4 r

2)r+ rrA(4 r2c

r) =0

Dividing by -4 r

8/13/2019 CRE II -27

8/17

Lets simply consider 1storder

0])/([ 22

rrdr

rdrdCDdcA

Ae

0])/([ 12

2

A

AeCkr

drrdrdCDd

c(-rA)=-rA volumetric

-rA=kCA

What about n-th order ?

0])/([ 2

2

n

An

AeCkr

dr

rdrdCDd Differentiation &Divide byr2De

8/13/2019 CRE II -27

9/17

1storder

What about n-th order ?

022

2

nA

e

nAA CD

k

dr

dC

rdr

Cd

02 1

2

2

A

e

AAC

D

k

dr

dC

rdr

Cd

8/13/2019 CRE II -27

10/17

Dimensionless Form of the Equation

Dimensionless symbol are normally introduced to

Reduce complexity in equation

Simplify operation of calculation

Scale-up the reactor

Let = CA/CAs and =r/R

dCA/dr= (dCA/d)(d/dr)= (d/d)(dCA/d )x(d/dr)

=> dCA/dr = (d/d)(CAS/R)d2C

A/dr2= d/dr(dC

A/dr)= (d2/d2)(C

AS/R2)

When

CA=CAsat r=R, => =1 and =1

CA=finite at r=0, => =finite and =0

8/13/2019 CRE II -27

11/17

8/13/2019 CRE II -27

12/17

Thiele Modulus, n

2 n-1 n

2 n As n Asn

e e As

k R C k RC "a" surface reaction rate = = =D D [(C -0)/R] "a" diffusion rate

If n is largeinternal diffusion limits the

overall rate

If n is smallthe surface reaction limits the

overall rate

8/13/2019 CRE II -27

13/17

13

Define y= => = y/

d /d = 1/ (dy/d ) y/2d2 /d 2 = 1/d2y/d2 - 2/2dy/d + 2y/ 3

d2y/d 2 - 2y=0 Solutiony= A Cosh + B Sinh

A=0 as must be finite at the centre

(B. C =0, cos h 1; 1/ , and Sin h 0.

A 1

As 1

C sinh 1 = =C sinh

8/13/2019 CRE II -27

14/17

14

The total rate of consumption Of A inside a pellet

Evaluated at the surface conditions

8/13/2019 CRE II -27

15/17

Internal Effectiveness Factor

InternaleffectivenessFactor, is:ranged 01

for a first-orderreaction in aspherical catalystpellet

As s

Actual overall rate of reaction = 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 -r = = =

-r -r -r

1 121

3 = coth -1

8/13/2019 CRE II -27

16/17

16

For STRONG PORE DIFFUSION

8/13/2019 CRE II -27

17/17