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L -28 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

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1storder Reaction in Spherical Pellet

What about n-th order ?

022

2

nA

e

nAA CDk

drdC

rdrCd

02 1

2

2

A

e

AAC

D

k

dr

dC

rdr

Cd

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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)d

2

CA/dr

2

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

2/d

2

)(CAS/R

2

)WhenCA=CAsat r=R, => =1 and =1

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

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Dimensionless eq.1storder

for n-th order ?

02

2

2

n

A

e

nAA CD

k

dr

dC

rdr

Cd

02 1

2

2

A

e

AAC

D

k

dr

dC

rdr

Cd 2

2

12

20

d d

d d

22

2

20

n

n

d d

d d

Thiele

Module

Thiele

Module

eDRk 21

e

n

Asn

D

CRk 12

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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

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6

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

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The total rate of consumption Of A inside a pellet

Evaluated at the surface conditions

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Internal Effectiveness Factor

As s

Actual overall rate of reaction =

Rate of reaction that would result if entire

interior surface were exposed to the externalpellet surface conditions C ,T

' "

A A A

' "

As As As

-r -r -r = = =

-r -r -r

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For STRONG PORE DIFFUSION

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Internal Effectiveness Factor

1 12

1

3

= coth -1

Internal effectivenessFactor, is: ranged 01

for a first-orderreaction in a spherical

catalyst pellet

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Calculation of Catalytic Effectiveness FactorCatalytic Effectiveness Factor:

where

- Thiele Modulus

1storder reaction rate:

Spherical Pellet

Cylindrical Pellet

Slab Pellet

)313(1

Coth

DekSaR p /3

DekSaR

p/2

DekSaL p /

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How can the rate of reaction be increased?

Decrease the radius of the pellet.

increase the temperature.

Increase the concentration

Increase the internal surface area

For large value of Thiele modulus, the effectiveness factor (for n>1

efffectiveness factor decrease with increase concentration at the

external surface of the pellet

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For nthorder reaction

= ( 2/n+1 )1/23/

= R( k CASn-1/ De)1/2

OR

2= R2kSap CASn-1/De

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Uses the measured values of the rate of reaction to

determine if Internal diffusion controls the rate.

Weisz-Prater Parameter CWP2= 3( Coth -1)

2 = (observed rate/rate cal. at CAS) x (rate

calculated at CAS) / diffusion Rate)

= (-rA(obs)/ -rAs

2= -rASSapR2/De CAs = -rAS pR

2/De CAs

WeiszPrater Criterion for internal diffusion

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CWP= (-rA(obs)/ -rAs )(-rASpR2/De CAS)

CWP = (-rA(obs) (pR2/De CAS)

These are measured or known terms.

if CWP > 1, Internal diffusion limits the rate.

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Falsified Kinetics Measurement of the apparentreaction order

and activation energy results primarily wheninternal diffusion limitations are present.

This becomes serious if the catalyst pelletshape and size between lab (apparent) andreal reactor (true) regime were Too different.

Smaller catalyst pelletreduces the diffusion

limitationhigher activation energy moretemperature sensitive

RUNAWAY REACTION CONDITIONS!!!!

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Falsified Kinetics

With the same rate of production, reaction order

and activation energy to be measured

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Determination of apparent order of reaction

For large value of Thiele modulus

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Apparent order of reaction n = (ntrue+1 )/2

N i th l ll t E b l

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Non isothermal pellet Energy balance

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Non isothermal pellet effectiveness factor

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Overall effectiveness factor (Both internal and external

diffusion are important

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Slove for Cas, Conc at pellet

Surface

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Overall rate in terms of

bulk concentration

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Rate of reaction, -rA

= (Actual overall rate of reaction) divided by(rate that would result if the entire surface

were exposed to the bulk conditions, CAb,Ts)

"

1 a b c c

" " "A Ab 1 Ab

' " "

A A b A a b 1 Ab a b

=1+k S /k a

-r = (-r ) = k C

-r = -r = -r S = k C S

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Mass transfer and reaction in a

packed bed

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'AzA b

AbAz AB Ab

2'Ab Ab

AB A b2

dW- +r = 0

dz

where

dCW = -D +C U and

dz

U = superficial velocity

hence

d C dCD -U +r = 0

dz dz

Mole balance in flux form, where Acis

constant and FA= AcWAz =ACU CA

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Mass Balance in a Packed bed (Mass Transfer

control)

FAzFAZ+Z - (-rA) a Ac z = 0

A= surface area per unit volume, U superfecial

velocity.

U dCA/ dz = - kcCA ac

. CA= CA0exp (- kcac z/U)

Or ln (1-X) = ( -kc az/U)

0.0)()( iSGiccGiG CCakdz

dCU

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Catalytic Fixed-Bed Reactor - Design Model

Mass Balance around the catalyst

Gas-Phase component mass balance (Plug Flow model)

Gas-Phase component mass balance (Dispersion model)

Energy Model

inetSGicc RiCCak )()()(

0.0)()( iSGiccGi

G CCakdz

dCU

0.0)()(2

2

iSGiccGiGGiG CCakdz

dCU

zd

CdiD

)()( TaTUAjHRjdzdTCpU RGGG

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Overall Rate with in the pellet (-rA= (rAb)

For first order reaction : (-rAb;= (rAb)Sa= ksSa CAb

=>(-rA)= (rAb) = Sabk CAb

2

2

0s b bAb Ab Ad C dC

Da U k SaCdz dz

Neglecting Axial Dispersion

( )s aAb

AbdC k S

Cdz U

=> CAb= CAb0 exp(- Sa bks Z/U)

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Axial diffusion, can be neglected when

FAis very large

so

Finally, the conversion for

1storder reaction in PBR is

'

0 p A b p

a 0 Ab

U d -r d>>

D U C

2Ab

a 2

d CDdz

"

Ab b aAb

dC k S= - C

dz U

Remember the

forced

convection in

binary external

diffusion, JAisalso neglected

b a-( k"S L)/UAb

Ab0

CX =1- =1- e

C

Mass transfer and reaction in a packed bed

cont.

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Determination of limiting situation from

reaction data

Type ofLimitation

Variation of Reaction Rate with:

Velocity

Particle

Size TemperatureExternaldiffusion

U (dp)-3/2

Rate= kcac CA

Linear

InternalDiffusion

Independent (dp)-1 Exponential

SurfaceDiffusion

Independent

Independent

Exponential