Internal Diffusion Effect

INTERNAL DIFFUSION EFFECTSINTERNAL DIFFUSION EFFECTS(8)(8)

Marcel LacroixMarcel LacroixUniversitUniversitéé de Sherbrookede Sherbrooke

INTERNAL DIFFUSION EFFECTS:INTERNAL DIFFUSION EFFECTS:INTRODUCTIONINTRODUCTION

• IN A HETEROGENEOUS REACTION SEQUENCE, MASS TRANSFER OF REACTANTS FIRST TAKES PLACE FROM THE BULK FLUID TO THE EXTERNAL SURFACE OF THE PELLET. THE REACTANTS THEN DIFFUSE FROM THE EXTERNAL SURFACE INTO AND THROUGH THE PORES WITHIN THE PELLET, WITH REACTION TAKING PLACE ONLY ON THE CATALYTIC SURFACE OF THE PORES.

M. Lacroix Internal Diffusion Effects 2

INTERNAL DIFFUSION EFFECTS:INTERNAL DIFFUSION EFFECTS:INTRODUCTIONINTRODUCTION

• IN THE PREVIOUS DISCUSSION OF SURFACE REACTIONS, WE ASSUMED THAT EACH POINT IN THE INTERIOR OF THE ENTIRE CATALYST SURFACE WAS ACCESSIBLE TO THE SAME REACTANT CONCENTRATION.

• HOWEVER, WHERE THE REACTANTS DIFFUSE INTO THE PORES WITHIN THE CATALYST PELLET, THE CONCENTRATION AT THE PORE MOUTH WILL BE HIGHER THAN THAT INSIDE THE PORE, AND WE SEE THAT THE ENTIRE CATALYTIC SURFACE IS NOT ACCESSIBLE TO THE SAME CONCENTRATION.


INTERNAL DIFFUSION EFFECTS:INTERNAL DIFFUSION EFFECTS:OBJECTIVEOBJECTIVE

• TO EXAMINE THE EFFECTS OF DIFFUSION OF THE REACTANTS INTO THE PORES WITHIN THE CATALYST PELLET ON THE OVERALL RATE OF REACTION.


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:MOLE BALANCE EQUATIONMOLE BALANCE EQUATION

• STEADY-STATE MOLE BALANCE ON SPECIES A AS IT ENTERS, LEAVES AND REACTS IN A SPHERICAL SHELL OF INNER RADIUS

AND OUTER RADIUS OF THE PELLET:rr ∆+r

( ) ( ) ( ) 0___ =+− ∆∆+ rrrr generatedAoutAinA



• SUBSTITUTION IN THE MOLE BALANCE EQUATION

YIELDS:

( ) 24_ rArr rWinA π⋅=

( ) 24_ rrArrr rWoutA ∆+∆+ ⋅= π

( ) rrrgeneratedA mcAr ∆=∆2' 4_ πρ

0)( 2'2

=− rrdr

rWdcA

Ar ρ

RATE OF REACTION PER MASS OF CATALYST (mole/s kg cat.)

DENSITY OF CATALYST (kg/m3)



• MOREOVER,

AND

• :EFFECTIVE DIFFUSIVITY (m2/s)• :RATE OF REACTION PER UNIT SURFACE AREA OF

CATALYST (mole/s m2)• :SURFACE AREA OF THE CATALYST PER UNIT MASS OF

CATALYST (m2/g cat.). TYPICAL VALUE: 150 m2/g cat. • :n-ORDER SPECIFIC REACTION RATE CONSTANT

(m3/mole)n-1(m/s)

drdCDW A

eAr −=a

nAnaAA SCkSrr =−=− '''

eD''

Ar

aS

nk


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFERENTIAL EQUATION DIFFERENTIAL EQUATION

• THUS, THE DIFFERENTIAL EQUATION DESCRIBING DIFFUSION AND REACTION WITHIN A SPHERICAL PELLET BECOMES:

WITH

• USING THE DIMENSIONLESS VARIABLES AND, IT BECOMES

WHERE

022

2

=−+ nA

e

canAA CDSk

drdC

rdrCd ρ ( )

( ) AsRrA

rA

CCfiniteC

=

=

=

= ;0

02 22

2

=−+ nnd

ddd ϕφ

λϕ

λλϕ

AsA CC /=ϕRr /=λ

e

nAscan

n DCRSk 12

2−

=ρφ


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:THIELE MODULUSTHIELE MODULUS

• IS THE THIELE MODULUS.

• IS A MEASURE OF THE RATIO OF A SURFACE REACTION RATE TO A RATE OF DIFFUSION THROUGH THE CATALYST PELLET.

• WHEN THE THIELE MODULUS IS LARGE, INTERNAL DIFFUSION USUALLY LIMITS THE OVERALL RATE OF REACTION; WHEN IT IS SMALL, THE SURFACE REACTION IS USUALLY RATE-LIMITING.

e

nAscan

n DCRSk 12

2−

=ρφ

nφ

2nφ


REACTION RATES: REMINDERREACTION RATES: REMINDER

• : RATE OF REACTION PER UNIT VOLUME (mole/m3s)

• : RATE OF REACTION PER UNIT MASS OF CATALYST (mole/g cat. s)

• : RATE OF REACTION PER UNIT SURFACE AREA OF CATALYST (mole/m2 s)

• : SURFACE AREA OF THE CATALYST PER UNIT MASS OF CATALYST (m2/g cat.). TYPICAL VALUE: 150 m2/g cat.

• THUS, AND FOR FIRST-ORDER REACTION,

''Ar

aS

'Ar

Ar

aAA Srr ''' −=−

AA Ckr 1'' =− k1 :specific reaction rate constant (m/s)


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:SOLUTION FOR A FIRSTSOLUTION FOR A FIRST--ORDER REACTION (ORDER REACTION (n=1n=1))

• THE SOLUTION TO THE DIFFERENTIAL EQUATION IS

WITHe

can

DRSk 2

21

ρφ =⎟⎠

⎞⎜⎝

⎛==

)sinh()sinh(1

1

1

φλφ

λϕ

As

A

CC

SMALL VALUES OF THE THIELE MODULUS INDICATE SURFACE REACTION CONTROLS AND A SIGNIFICANT AMOUNT OF THE REACTANT DIFFUSES WELL INTO THE PELLET INTERIOR WITHOUT REACTING. LARGE VALUES OF THE THIELE MODULUS INDICATE THAT THE SURFACE REACTION IS RAPID AND THAT THE REACTANT IS CONSUMED VERY CLOSE TO THE EXTERNAL PELLET SURFACE AND VERY LITTLE PENETRATES INTO THE INTERIOR OF THE PELLET.


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:INTERNAL EFFECTIVENESS FACTORINTERNAL EFFECTIVENESS FACTOR

• THE INTERNAL EFFECTIVENESS FACTOR IS THE RATIO OF THE ACTUAL OVERALL RATE OF REACTION TO THE RATE OF REACTION THAT WOULD RESULT IF THE ENTIRE INTERIOR SURFACE WERE EXPOSED TO THE EXTERNAL PELLET SURFACE CONDITIONS CAs AND Ts

• TO DERIVE THE EFFECTIVENESS FACTOR FOR A FIRST-ORDER REACTION, LET US WORK WITH REACTION RATES IN MOLES PER UNIT TIME:

''

''

'

'

As

A

As

A

As

A

rr

rr

rr

−−

=−−

=−−

=η

η

As

A

As

A

As

A

MM

catalystmassrcatalystmassr

rr

=⋅−⋅−

=−−

=)_()_(

'

'

'

'

η

)10( <<η


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:INTERNAL EFFECTIVENESS FACTORINTERNAL EFFECTIVENESS FACTOR

• IF THE ENTIRE SURFACE WERE EXPOSED TO THE CONCENTRATION AT THE EXTERNAL SURFACE OF THE PELLET, CAs THE RATE FOR A FIRST-ORDER REACTION WOULD BE:

• THE ACTUAL RATE OF REACTION IS THE RATE AT WHICH THE REACTANT DIFFUSES INTO THE PELLET AT THE OUTER SURFACE:

• THUS, THE FFECTIVENESS IS

caAscAsAs RSCkRrM ρπρπ 31

3'

34

34

⋅=⋅−=

)1)coth((4)1(4 11 −=== φφπλλϕπ AseAseA CRD

ddCRDM

THE SUBSCRIPT s INDICATES THAT THE RATE IS EVALUATED AT THE CONDITIONS PRESENT AT THE EXTERNAL SURFACE OF THE PELLET.

)1)coth((3112

1

−== φφφ

ηAs

A

MM

(8.1)


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:RATE OF REACTIONRATE OF REACTION

• FOR ,

• AND FOR A FIRST-ORDER REACTION,

• THUS TO INCREASE THE RATE OF REACTION, (1) DECREASE THE RADIUS R; (2) INCREASE THE TEMPERATURE; (3) INCREASE THE CONCENTRATION; AND (4) INCREASE THE INTERNAL SURFACE AREA.

21

1112

1

33)1(3⎟⎟⎠

⎞⎜⎜⎝

⎛=≈−≈

ca

e

SkD

R ρφφ

φη

21 >φ

Asc

aeaAsAsA CkSD

RSCkrr

21

11

'' 3))(()( ⎟⎟⎠

⎞⎜⎜⎝

⎛==−=−

ρηη


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:INTERNAL EFFECTIVENESS FACTOR INTERNAL EFFECTIVENESS FACTOR

AS THE PARTICLE DIAMETER BECOMES VERY SMALL, THE THIELE MODULUS DECREASES, SO THAT THE EFFECTIVENESS FACTOR APPROACHES 1 AND THE REACTION IS SURFACE-REACTION-LIMITED. ON THE OTHER HAND, WHEN THE THIELE MODULUS IS LARGE (>30), THE INTERNAL EFFECTIVENESS FACTOR IS SMALL AND THE REACTION IS DIFFUSION-LIMITED WITHIN THE PELLET.


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:OVERALL EFFECTIVENESS FACTOROVERALL EFFECTIVENESS FACTOR

• WE NOW CONSIDER A SITUATION WHERE EXTERNAL AND INTERNAL RESISTANCE TO MASS TRANSFER TO AND WITHIN THE PELLET ARE OF THE SAME ORDER OF MAGNITUDE.



• AT STEADY-STATE, THE TRANSPORT OF THE REACTANT(S) FROM THE BULK FLUID TO THE EXTERNAL SURFACE OF THE CATALYST IS EQUAL TO THE NET RATE OF REACTION OF THE REACTANT WITHIN THE PELLET.

• THE MOLAR RATE OF MASS TRANSFER FROM THE BULK FLUID TO THE EXTERNAL SURFACE IS

VaWM cAA ∆⋅=

MOLAR FLUX (mole/m2s)

SURFACE AREA PER UNIT REACTOR VOLUME

REACTOR VOLUME

(8.2)



• THE NET (TOTAL) RATE OF REACTION ON AND WITHIN THE PELLET IS:

• COMBINING (8.2) AND (8.3) AND TAKING INTO ACCOUNT THAT FOR MOST CATALYSTS THE INTERNAL SURFACE AREA IS MUCH GREATER THAN THE EXTERNAL SURFACE AREA,

VSrVarM caAcAA ∆−−∆−= )1('''' φρ

EXTERNAL AREA INTERNAL AREA

)1('' φρ −−= caAcA SraW

(8.3)

(8.4)



• THE RATE OF MASS TRANSPORT IS ALSO GIVEN IN TERMS OF THE MASS TRANSFER COEFFICIENT :

• ASSUMING THAT THE SURFACE REACTION IS FIRST-ORDER WITH RESPECT TO A, WE CAN UTILIZE THE INTERNAL EFFECTIVENESS FACTOR TO WRITE

• COMBINING EQUATIONS (8.4)-(8.6) TO ELIMINATE CAs WE OBAIN THE MEASURABLE NET RATE OF REACTION

(8.5)

(8.6)

ck

VaCCkVaWM cAsAbccAA ∆−=∆⋅= )(

AsA Ckr 1'' η=−

Abccca

A CkakSk

r 11

''

)/())1((1 φρηη−+

=− (8.7)



• CONSEQUENTLY, THE OVERALL RATE OF REACTION IN TERMS OF THE BULK CONCENTRATION CAb IS

WHERE (8.8)

AbAA Ckrr 1'''' )( Ω=−⋅Ω=−

)/())1((1 1 ccca akSk φρηη−+

=Ω

INTERNAL EFFECTIVENESS

OVERALL EFFECTIVENESS


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:WEISZWEISZ--PRATER CRITERION FOR INTERNAL DIFFUSIONPRATER CRITERION FOR INTERNAL DIFFUSION

• THE WEISZ-PRATER CRITERION USES MEASURED VALUES OF THE RATE OF REACTION TO DETERMINE IF INTERNALDIFFUSION IS LIMITING THE REACTION.

• THIS CRITERION IS DEVELOPED INTUITIVELY FROM THE INTERNAL EFFECTIVENESS FACTOR:

• THE LEFT-HAND SIDE IS THE WEISZ-PRATER PARAMETER

• IF THERE ARE NO DIFFUSION LIMITATIONS AND CONSEQUENTLY NO CONCENTRATION GRADIENT EXISTS WITHIN THE PELLET.

)(' obsrA−

)1coth(3 112

1 −= φφηφ

Ase

cAWP CD

RobsrC2'

21

)( ρηφ −==

1<<WPC


EXAMPLE No. 1:EXAMPLE No. 1:DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:

WEISZWEISZ--PRATER CRITERIONPRATER CRITERION

• THE FIRST-ORDER REACTION WAS CARRIED OUT OVER TWO DIFFERENT-SIZED PELLETS. THE PELLETS WERE CONTAINED IN A SPINNING BASKET REACTOR THAT WAS OPERATED AT SUFFICIENTLY HIGH ROTATION SPEEDS THAT EXTERNAL MASS TRANSFER RESISTANCE WAS NEGLIGIBLE. THE RESULTS OF TWO EXPERIMENTAL RUNS MADE UNDER IDENTICAL CONDITIONSARE AS GIVEN IN THE FOLLOWING TABLE. ESTIMATE THE THIELE MODULUSAND EFFECTIVENESS FACTOR FOR EACH PELLET. HOW SMALL SHOULD THE PELLETS BE MADE TO VITUALLY ELIMINATE ALL INTERNAL DIFFUSION RESISTANCE?

BA→

MEASURED RATE(mole/g cat. s) X 105

PELLET RADIUS(m)

RUN 1 3.0 0.01

RUN 2 15.0 0.001


DIFFUSION AND REACTION IN SPHERICAL PELLETS:DIFFUSION AND REACTION IN SPHERICAL PELLETS:MEARSMEARS’’ CRITERION FOR EXTERNAL DIFFUSIONCRITERION FOR EXTERNAL DIFFUSION

• THE MEARS’ CRITERION USES THE MEASURED RATE OF REACTION TO LEARN IF MASS TRANSFER FROM THE BULK GAS PHASE TO THE CATALYST SURFACE CAN BE NEGLECTED.

• MASS TRANSFER EFFECTS CAN BE NEGLECTED IF

• :REACTION ORDER• :CATALYST PARTICLE RADIUS (m)• :BULK CONCENTRATION (kmole/m3)• :MASS TRANSFER COEFFICIENT (m/s)• :BED POROSITY

)(' obsrA−

15.0)1('

<−−

Abc

cA

CkRnr ρφ

nR

AbCckφ


MASS TRANSFER AND REACTION IN A PACKED BEDMASS TRANSFER AND REACTION IN A PACKED BED

• WE NOW CONSIDER THE ISOMERIZATION REACTION TAKING PLACE IN A PACKED BED OF CATALYSTS PELLETS:

• :CROSS-SECTIONAL AREA OF THE TUBE (dm2);• :BULK GAS CONCENTRATION OF A (mole/dm3);• :BULK DENSITY OF THE CATALYST BED ( , kg/m3);• :VOLUMETRIC FLOW RATE (dm3/s);• :SUPERFICIAL VELOCITY ( ,dm/s)

cρφ )1( −=

cAAbCbρ

cAv /0=U0v

BA→



( ) ( ) ( ) 0___ =+− ∆+ generatedAoutAinA zzz

( ) czAzr AWinA ⋅= )(_WHERE

( ) czzAzzr AWoutA ⋅= ∆+∆+ )(_

( ) zArgeneratedA cbA ∆= ρ'_



• THE MOLE BALANCE EQUATION BECOMES

• THE MOLAR FLUX IS THE SUM OF MASS DIFFUSION AND CONVECTION, i.e.,

• THE RATE OF REACTION

AbAb

ABAz UCdz

dCDW +−=

0' =+− bAAz r

dzdW ρ (8.9)

(8.10)

Ω=Ω−=Ω−=− aAbaAbAbA SkCSrrr '''' (8.11)

FIRST-ORDER REACTION



• SUBSTITUTION OF (8.10) AND (8.11) IN (8.9) YIELDS

• IF THE FLOW RATE THROUGH THE BED IS VERY LARGE, AXIAL DIFFUSION CAN BE NEGLECTED

AND THE SOLUTION TO (8.12) BECOMES FOR CONVERSION X

02

2

=Ω−− AbabAbAb

AB CkSdz

dCUdzCdD ρ

dzdCU

dzCdD AbAb

a <<2

2

(8.12)

)/exp(110

ULkSCCX ab

Ab

Ab Ω−−=−= ρ



• YOUNG AND FINLAYSON (Ind. Eng. Chem. Fund., 12, 412 (1973)) HAVESHOWN THAT AXIAL DISPERSION CAN BE NEGLECTED WHEN

• :SUPERFICIAL VELOCITY• :PARTICLE DIAMETER• :EFFECTIVE AXIAL DISPERSION COEFFICIENT

a

p

Ab

pbA

DdU

CUdr 0

0

'

<<− ρ

0UpdaD


EXAMPLE No. 2:EXAMPLE No. 2:MASS TRANSFER AND REACTION IN A PACKED BEDMASS TRANSFER AND REACTION IN A PACKED BED

• IT IS PROPOSED TO REDUCE THE CONCENTRATION OF NO IN AN EFFLENT STREAM FROM A PLANT BY PASSING IT THROUGH A PACLED BED OF SPHERICAL POROUS CARBONACEOUS SOLID PELLETS. A 2% NO – 98% AIR MIXTURE FLOWS AT A RATE OF 10-6 m3/s THROUGH A TUBE OF CROSS SECTION AREA OF 2.03 x 10-3

m2 PACKED WITH POROUS SOLID AT A TEMPERATURE OF 1173 K AND A PRESSURE OF 101.3 kPa. THE REACTION

IS FIRST-ORDER IN NO, THAT IS,

AND OCCURS PRIMARILY IN THE PORES INSIDE THE PELLET, WHEREAND

CALCULATE THE MASS OF POROUS SOLID NECESSARY TO REDUCE THE NO CONCENTRATION TO A LEVEL OF 0.004%, WHICH IS BELOW THE ENVIRONMENTAL PROTECTION AGENCY LIMIT.

221 NCOCNO +→+

NOaNO CkSr =− '

gmSa /530 2= smmk 2310 /1042.4 −⋅=


EXAMPLE No. 2:EXAMPLE No. 2:MASS TRANSFER AND REACTION IN A PACKED BEDMASS TRANSFER AND REACTION IN A PACKED BED

• AT 1173 K, THE FLUID PROPERTIES ARE:

• THE PROPERTIES OF THE CATALYST AND BED ARE:

;/1082.1

;/100.2

;/1053.1

28

28

28

smD

smD

sm

e

AB−

−

−

⋅=

⋅=

⋅=υ

;0.1;103

;/104.1)1(

;5.0;/108.2__/8.2

3

36

363

=⋅=

⋅=−=

=⋅=

−

γ

φρρ

φρ

mR

mg

mgorcmg

cb

c


LIMITING SITUATIONS FROM REACTION DATALIMITING SITUATIONS FROM REACTION DATA

• FOR EXTERNAL MASS TRANSFER-LIMITED REACTIONS IN PACKED BEDS, THE RATE OF REACTION AT A POINT IN THE BED IS

• WHEN INTERNAL DIFFUSION LIMITS THE RATE OF REACTION, WE OBSERVE THAT THE RATE OF REACTION VARIES INVERSELY WITH PARTICLE DIAMETER AND IS INDEPENDENT OF VELOCITY.

AccA Cakr =−

MASS TRANSFER COEFFICIENT

EXTERNAL SURFACE AREA PER UNIT REACTOR VOLUME

(mole/dm3s)

21

21

p

cd

Uk ∝p

c da 1∝

231

p

Ad

r ∝−


LIMITING SITUATIONS FROM REACTION DATALIMITING SITUATIONS FROM REACTION DATA

VARIATION OF REACTION RATE WITH:

TYPE OF LIMITATION

VELOCITY PARTICLE SIZE

TEMPERATURE

EXTERNAL DIFFUSION

LINEAR

INTERNAL DIFFUSION

INDEPENDENT EXPONENTIAL

SURFACE REACTION

INDEPENDENT INDEPENDENT EXPONENTIAL

21

U 23−

pd

1−pd


Internal Diffusion Effect

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internal surface

internal diffusion

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