topic 3.2- internal diffusion and reaction

8/10/2019 Topic 3.2- Internal Diffusion and Reaction

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CHE 625

TOPIC 3.2 : Diffusion and

ReactionRef: Fogler (Chapter 12) 4th ed.

Pg 813 onwards



SUBTOPICS

Diffusion and Reaction in Spherical Catalyst PelletsEffectiveness factorEstimation of diffusion and Reaction limited regimesPore diffusion Resistance and Surface kineticsPorous catalyst particles

Gas liquid reaction on solid catalyst reaction and itsapplication



CHEE 32322.3

Internal Mass Transfer in Porous Catalysts

Previously, we have examined the potential influence of external mass transferon the rate of heterogeneous reactions.However, where active sites are accessible within the particle, internal masstransfer (molecular diffusion) has a tremendous influence on the rate ofreaction within the catalyst. Numerous examples exist:

Encapsulated or entrapped enzymesMicroporous catalysts for catalytic cracking (zeolites)

The diffusion rate of reactants and products within the particle oftendetermines the rate at which a microporous catalyst functions.



CHE625

Concentration Gradients of Diffusing Reactants and ProductsIn a Uniform Catalyst Particle

Spherical catalyst particle with diffusingreactant. C As is the concentration ofreactant at the particle periphery.

CAs

* Concentration at R (pore mouth) should behigher as compared to concentration at anypoint inside the pore



22.5

Intraparticle Mass Transport

A simple but conceptually useful treatment of intra-particlemass transfer describes the diffusion of a reactant within auniformcatalyst particle.

Assumptions:Uniform catalytic activity throughout the spherical particleUniform properties of solid

Irreversible, first-order kinetics; rate = k C A

Given that mass transfer occurs by molecular diffusion, ananalytical expression for the transport and consumption ofour reactant can be written.

CAs



Pores are tortuous and have varying cross-sectional areaNeed to define an effective diffusion coefficient in radialdirectionEffective Diffusivity (De) is a measure of diffusivity thataccounts for the following:

Not all area normal to flux direction is available for molecules todiffuse in a porous particle ( P)

Diffusion paths are tortuous ( )Pore cross-sections vary ( )Internal void fraction, s = P

Go thru’ Example 12 -1 pg 815 (Fogler)

Diffusion/Rxn in Porous Catalysts

~DD P A

e



1. Perform Shell Balance in the radial direction and outward(increasing r) - in terms of molar flux times area

2. Write the expression for WAr (for EMCD and dilute

concentrations )3. Write the corresponding rate law (relate –r A, -r A’ and -r A”) and

relate with shell balance4. Set boundary conditions for C A (based on position r )

5. Write the equation in dimensionless form to obtain Φ (Thielemodulus) in terms of two dimensionless quantities ψ and λ

6. Obtain W expression with dimensionless quantities included7. Determine concentration profile for the spherical cat. pellet.

STEPS IN INTERNAL DIFFUSION &REACTION




steady state mass balance

rate in at r

r

2 Ar r 4W

rate out at r + r

r r

2 Ar r 4W

rate of generation within shell

c

mass catalystrate reaction

volume shellmass catalyst

volume shell

rr + r

R

r r 4 2m

'

Ar

0r r 4r

r 4Wr 4W

2mC

' A

r r

2 Ar r

2 Ar

0r r dr

r Wd 2C

' A

2 Ar

B A cat

Step #1 : Perform Shell balance in radial direction




0r r dr

r Wd 2C

' A

2 Ar

dr dC

DW Ae Ar

0r r r dr dCDdr d 2C' A2 Ae

0r Sr r dr

dCD

dr

d 2

Ca

"

A

2 Ae

2 AnaC A

2

Ana

'

A

2

An

"

A

CkSr

CkSr

Ckr

0r SCk 2Ca

n An

rate equationdefinitions

substitute Fick

s Law

Step 3: Write rate laws

Step 2: Write the expression for W Ar




0r SCkr dr

dCDdr d 2

Can An

2 Ae

finiteC 0r A symmetry

AsRr A CC surface

Step 5a: Write the equationin dimensionless form

As

A

CC

Rr

As A C

ddC

R1

ddr

RC

dd

dr dC As A

0CDSk

dr dC

r 2

dr Cd n

Ae

Can A2 A

2

0D

CRSkdd2

dd n

e

1n As

2Can

2

2

Step 4: Identify boundary conditions




Step 5b : define Thiele modulus (n)

0D

CRSkdd2

dd n

e

1n As

2Can

2

2

e

1n

As

2

Can2n D

CRSk

0dd2

dd n2

n2

2

understand the Thiele modulus

R0CDRCSk

Ase

n AsCan2

nr eac t ion r a t e

d i f f u s i o n r a t e

large n - diffusion controls

small n - kinetics control

0CD

Sk

dr

dC

r

2

dr

Cd n A

e

Can A

2

A2



Diffusion/Rxn in Porous Catalystsfirst orderkinetics(n = 1)

define y =

0dd2

dd 2

12

22

e

Can21 R

DSk

322

2

2

2 y2ddy2

dyd1

dd

2

y

d

dy1

d

d

0yd

yd 212

2

1111 sinhBcosh Ay

1B

1 A sinhcosh 11

differential has the solution apply boundary conditions

1 ,1

finiteis ,0



Diffusion/Rxn in Porous Catalystsfirst orderkinetics(n = 1)

0dd2

dd 2

12

2 2

e

Can21 R

DSk

0yd

yd 212

2

1111 sinhBcosh Ay

1B

1 A sinhcosh 11

differential has the solution apply boundary conditions

1 ,1

finiteis ,0

1

1

sinhsinh1

As

A

CC



Thiele Modulus

As

A

CC

Refer Figure 12-4 pg 823for concentrationprofile

d 1 1 sinh 1 sinh 1

00.510

0.5

1

1

2 5 20

Go thru’ Example 12 -2 pg823



The internal effectiveness factor ( ) is a measure of therelative importance of diffusion to reaction limitations :

Internal Effectiveness Factor ( )

s As T,Ctoexposedweresurfaceentireif rateratereactionoverallactual

As

A"

As

" A

' As

' A

As

A

MM

r r

r r

r r M mol / time

r mol / time / mass cat




Determine M As (rate if all surface at C As)

catalystmass

catalystmass

areasurfaceareaunitper rateM As

' Asr

aS

CV AsM

x

x

As1Ck

c3

34

a As1 As RSCkM



Determine M A (actual rate is equal to reactant diffusionrate at outer surface)


1 Ase A d

dCRD4M

11

12

1

11

1 sinhsinh1

sinhcosh

dd

1coth 11

1cothCRD4M 11 Ase A



Substitute results into definition of


As

A

MM

c

3

3

4

a As1

11 Ase

RSCk1cothCRD4

1cothRSk

D3 11

c2

a1

e

1coth3112

1




1coth3

1121

0.1 1 10 1000.1

1

12

131

21

101

ac1

e21

SkD

R33

1 20

small d p




1coth3

1121

0.1 1 10 1000.1

1

12

131

21

101

ac1

e21

SkD

R33

1 20

reactionrate

limited

internaldiffusionlimited



22.21

Effectiveness Factor

Note that internal diffusion resistance decreases with decreasing. Therefore, the influence of diffusion on the reaction rate

supported by a particle is reduced when particle radius is reduced,DAB is high and the rate constant is relatively small.



Thiele modulus - Derived for spherical particle geometry

Derived for 1 st order kineticsFor large , approximately

Internal effectiveness factor - Assumed =0, correction applied when 0Assumed isothermal conditions

Revisit and

21

213

1n2



For exothermic reactions, can be > 1 as internaltemperature can exceed T s.The rate internally is thus larger than at the surface

conditions where is evaluated.The magnitude of this effect is dependent on

Hrxn , Ts, Tmax , and k t (thermal conductivity of the pellet)

and are used to quantify this effect:

can result in mulitple steady statesNo multiple steady states exist if Luss criterion is fulfilled

Non-Isothermal Behavior

Number ArrheniussRT

Est

Aserxn

s

smax

TkCDH

TTT

14



When both internal AND external diffusion resistances areimportant (i.e., the same order of magnitude), both must be

accounted for when quantifying kinetics.It is desired to express the kinetics in terms of the bulkconditions, rather than surface conditions:

Overall Effectiveness Factor

bulk A,Ctoexposedweresurfaceentireif rateratereactionoverallactual



Accounting for reaction both on and within the pellet, themolar rate becomes:

For most catalyst, internal surface area is significantly higherthan the external surface area:


V1Sar M

cac

"

A A

b

bac"

Ac A Sar aW

ba"

Ac A Sr aW




ba"

Ac A Sr aW reaction rate(internal & external surfaces)

VaCCkVaW c Asbulk, Acc Ar mass transport rate

internal surfaces notall exposed to C As

As1"

As"

A Ckr r Relation between C As and C A defined by the as:

VSCkVaW ba As1c A

ba As1c Asbulk, Ac SCkaCCk




ba"



As1"

As"

A Ckr r Relation between C As and C A defined by the as:

ba1cc

bulk, Acc As Skka

CkaCSolving for C As :




ba"



ba1cc

bulk, Acc1" A

Skka

Cakkr Substitution into the rate law:

ba1cc

bulk, Acc As Skka

CkaCSolving for C As :




summary of factor relationships:

ba1cc

bulk, Acc1" A Skka

Cakkr

Rearranging the expression:

bulk, A1ccba1

CkakSk1

"bulk, A

" A r r ccba1 akSk1

" As

"bulk, A

" A r r r

As1"

As Ckr

Ab1"

Ab Ckr

Overall Effectiveness Factor ( )



Weisz-Prater Criterion is a method of determining if a givenprocess is operating in a diffusion- or reaction-limited regime

CWP is the known as the Weisz-Prater parameter. All quantities are

known or measured.

CWP << 1, no C in the pellet (kinetically limited)CWP >> 1, severe diffusion limitations

Weisz-Prater Criterion

Ase

c2'

obs, A21WP CD

Rr C

Go thru’ Example 12 -3 pg 839



Mass transfer effects negligible when it is true that

n is the reaction order, and the transfer coefficients k c and h(below) can be estimated from an appropriate correlation(i.e., Thoenes-Kramers for packed bed flow)

Heat transfer effects negligible when it is true that

Mears’

Criterion

15.0

Ck

nRr

Abc

b'

A

15.0ThR

REr H2

bg

b'

Arxn



0r dz

dCU

dzCd

D A'

A Ab

2 Ab

2

AB

Shell balance onvolume element A z

Mole flux of A

First order reaction

Application to PBRs – Mass Transferand Reaction

0r dz

dWb

' A

Az

UCdz

dCDW Ab A

AB Az

Aba'

Ab'

A CkSr r

0CkS Abab



0CkSdz

dCU

dzCd

D Abab Ab

2 Ab

2

AB

Axial dispersion negligible(relative to forced axialconvection) when…

dp is the particle diameterUo is the superficial velocity of the gasDa is the effective axial dispersion coefficient

Application to PBRs

a

po

Abo

pb'

A

D

dU

CU

dr

Which can be rewritten as:



Application to PBRs

Which can be rewritten as:

Abab Ab C

UkS

dzdC

Entrance condition:o Ab0z Ab CC

Integrating and applying boundary condtion yields:

UzkS

expCC ab Ab Ab o

Go thru’ Example 12 -4 pg 845



Problem in Class (pg 860-Fogler)



THE END of CHAPTER 3

topic 3.2- internal diffusion and reaction

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