EXTERNAL DIFFUSION EFFECTS (7) Marcel Lacroix Université ...

EXTERNAL DIFFUSION EFFECTSEXTERNAL DIFFUSION EFFECTS(7)(7)

Marcel LacroixMarcel LacroixUniversitUniversitéé de Sherbrookede Sherbrooke

EXTERNAL DIFFUSION EFFECTS:EXTERNAL DIFFUSION EFFECTS:OBJECTIVEOBJECTIVE

• TO EXAMINE THE EFFECT OF DIFFUSION OF THE REACTANTS OR PRODUCTS BETWEEN THE BULK FLUID AND EXTERNAL SURFACE OF THE CATALYST ON THE OVERALL RATE OF REACTION.

M. Lacroix External Diffusion Effects 2

ANALOGY BETWEEN HEAT AND MASS TRANSFER:ANALOGY BETWEEN HEAT AND MASS TRANSFER:FLOW AROUND A CATALYST PELLET FLOW AROUND A CATALYST PELLET

sT


EXTERNAL RESISTANCE TO HEAT TRANSFEREXTERNAL RESISTANCE TO HEAT TRANSFER

• HEAT FLUX FORM A BULK FLUID AT TO THE SURFACE OF A SPHERICAL PARTICLE OF DIAMETER AT : NEWTON’S LAW OF COOLING

• CALCULATION OF THE HEAT TRANSFER COEFFICENT:

• NUSSELT NUMBER:

∞TST

)( STThq −= ∞

Heat flux (Watt/m2) Heat transfer coefficient (Watt/m2K)

31

21

PrRe6.00.2 ⋅+=⋅

=t

p

kdh

Nu

µρ pdV∞=Re

tt

P

kC

αυµ

==Pr

pd

(REYNOLDS NUMBER) (PRANDTL NUMBER)


EXTERNAL RESISTANCE TO HEAT TRANSFEREXTERNAL RESISTANCE TO HEAT TRANSFER

• : FLOW VELOCITY (m/s);• : FLUID DENSITY (kg/m3);• : FLUID DYNAMIC VISCOSITY (Ns/m2);• : FLUID HEAT CAPACITY (J/kgK);• : FLUID THERMAL CONDUCTIVITY (W/mK);

• : FLUID KINEMATIC VISCOSITY (m2/s);

• : FLUID THERMAL DIFFUSIVITY (m2/s);

∞Vρµ

PC

tk

ρµυ =


PCt

tkα =ρ

EXTERNAL RESISTANCE TO MASS TRANSFEREXTERNAL RESISTANCE TO MASS TRANSFER

• MOLAR FLUX FORM A BULK FLUID AT CONCENTRATION TO THE SURFACE OF A SPHERICAL PARTICLE OF DIAMETER AT CONCENTRATION :

• CALCULATION OF THE MASS TRANSFER COEFFICENT:

• SHERWOOD NUMBER:

• REYNOLDS NUMBER:

• SCHMIDT NUMBER:

AbC

ASC)( ASAbcA CCkW −=

Molar flux (moles/m2s) Mass transfer coefficient (m/s)

31

21

Re6.00.2 ScD

dkSh

AB

pc ⋅+=⋅

=

µρ pdV∞=Re

ABDSc υ

=

pd


EXTERNAL RESISTANCE TO MASS TRANSFEREXTERNAL RESISTANCE TO MASS TRANSFER

• : FLOW VELOCITY (m/s);• : FLUID DENSITY (kg/m3);• : FLUID DYNAMIC VISCOSITY (Ns/m2);

• : FLUID KINEMATIC VISCOSITY (m2/s);

• : FLUID MASS DIFFUSIVITY (m2/s);• : BULK FLUID MOLAR CONCENTRATION

OF REACTANT OR PRODUCT A (mole/m3);

∞Vρµ

ρµυ =

ABD

AbC


DIFFUSIVITY RELATIONSHIPS FOR GASES, DIFFUSIVITY RELATIONSHIPS FOR GASES, LIQUIDS AND SOLIDSLIQUIDS AND SOLIDS


EXAMPLE No. 1:EXAMPLE No. 1:RAPID REACTION ON A CATALYST SURFACERAPID REACTION ON A CATALYST SURFACE

• CALCULATE THE MASS FLUX OF A REACTANT A TO A SINGLE CATALYST PELLET 1 CM IN DIAMETER SUSPENDESD IN A LARGE BODY OF LIQUID. THE REACTANT IS PRESENT IN DILUTE CONCENTRATIONS, AND THE REACTION IS CONSIDERED TO TAKE PLACE INSTANTANEOUSLY AT THE EXTERNAL SURFACE (i.e. CAs = 0). THE BULK CONCENTRATION OF THE REACTANT IS 1.0 M (1 M = 1.0 mole/dm3) , AND THE FREE-SYSTEM VELOCITY IS 0.1 m/s. THE KINEMATIC VISCOSITY IS 0.5 CENTISTOKE (1 CENTISTOKE=10-6m2/s), AND THE LIQUID DIFFUSIVITY OF A IS 10-10m2/s.


REACTION ON A CATALYST SURFACEREACTION ON A CATALYST SURFACE

• CONSIDER THE FOLLOWING ISOMERIZATION TAKING PLACE ON THE SURFACE OF A SOLID SPHERE: BA→

DIFFUSION TO, AND REACTION ON, EXTERNAL SURFACE OF PELLET

IF THE TEMPERATURE IS SUFFICIENTLY HIGH, THERE IS WEAK ADSORPTION AND THE SURFACE REACTION MAY BE EXPRESSED AS:

ASrAS Ckr =− ''

SPECIFIC REACTION RATE (m/s)


REACTION ON A CATALYST SURFACE: REACTION ON A CATALYST SURFACE: BOUNDARY CONDITIONSBOUNDARY CONDITIONS

• AT THE SURFACE OF THE PELLET ( ), MOLAR FLUX = RATE OF REACTION, i.e.,

• AT THE LIMIT OF THE BOUNDARY LAYER , MOLAR FLUX = CONVECTIVE TRANSPORT ACROSS THE BOUNDARY LAYER, i.e.,

)()( '' surfacersurfaceW AA −=

ASrASAbcA CkCCkboundaryW =−= )()(

0=y

δ=y


REACTION ON A CATALYST SURFACE: REACTION ON A CATALYST SURFACE: BOUNDARY CONDITIONSBOUNDARY CONDITIONS

• ELIMINATING (WHICH IS DIFFICULT TO MEASURE), THE RATE OF REACTION ON THE SURFACE BECOMES

• THE FLUX TO OR FROM THE SURFACE MAY BE WRITTEN IN TERMS OF AN EFFECTIVE TRANSPORT COEFFICIENT:

WHERE

ASC

cr

ArcASA kk

CkkrW+

=−= ''

AeffASA CkrW =−= ''

rc

rceff kk

kkk+

=


REACTION ON A CATALYST SURFACE: REACTION ON A CATALYST SURFACE: RAPID REACTIONRAPID REACTION

• THE RATE OF MASS TRANSFER TO THE SURFACE LIMITS THE OVERALL RATE OF REACTION, i.e.,

AND CONSEQUENTLY

• TO INCREASE THE RATE OF REACTION, ONE MUST INCREASE AND/OR

• THE MASS TRANSFER COEFFICIENT IS GIVEN BY:

cr kk >> AcAS Ckr ≈− ''

AC ck

21

21

61

32

31

21

6.06.00.2p

AB

AB

p

p

ABc

d

VDD

dVdDk ∞∞ ≈⎟

⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=

υ

υυ


REACTION ON A CATALYST SURFACE: REACTION ON A CATALYST SURFACE: RAPID REACTIONRAPID REACTION

• TO INCREASE THE MASS TRANSFER COEFFICIENT AND THUS THE OVERALL RATE OF REACTION PER UNIT SURFACE, ONE MAY EITHER DECREASE THE PARTICLE SIZE OR INCREASE THE VELOCITY OF THE FLUID FLOWING PAST THE PARTICLE.

• FOR EXAMPLE, IF THE VELOCITY IS DOUBLED, THE MASS TRANSFER COEFFICIENT AND CONSEQUENTLY THE RATE OF REACTION IS INCRESED BY A FACTOR OF

41.12 212

1

1

2 ==⎟⎠

⎞⎜⎝

⎛VV


REACTION ON A CATALYST SURFACE: REACTION ON A CATALYST SURFACE: SLOW REACTIONSLOW REACTION

• THE SPECIFIC REACTION RATE IS SMALL WITH RESPECT TO THE MASS TRANSFER COEFFICIENT, i.e., AND THUS

• THE SPECIFIC REACTION RATE IS INDEPENDENT OF THE VELOCITY OF THE FLUID AND FOR THE SOLID SPHERE CONSIDERED HERE, INDEPENDENT OF PARTICLE SIZE. HOWEVER, FOR POROUS CATALYSTS PELLETS, MAY DEPEND ON PARTICLE SIZE FOR CETAIN SITUATIONS.

cr kk << ArAS Ckr ≈− ''

rk


REACTION ON A CATALYST SURFACE: REACTION ON A CATALYST SURFACE: SLOW REACTIONSLOW REACTION

AT LOW VELOCITIES THE MASS TRANSFER BOUNDARY LAYER THICKNESS IS LARGE AND DIFFUSION LIMITS THE REACTION.

AT HIGH VELOCITIES, THE BOUNDARY LAYER THICKNESS IS SMALLER AND THE MASS TRANSFER ACROSS THIS LAYER NO LONGER LIMITS THE REACTION RATE. ALSO, FOR A GIVEN VELOCITY, REACTION-LIMITING CONDITIONS CAN BE ACHIEVED BY USING VERY SMALL PARTICLES. HOWEVER, THE SMALLER THE PARTICLE SIZE, THE GREATER THE PRESSURE DROP IN A PACKED BED.


MASS TRANSFERMASS TRANSFER--LIMITED REACTIONS IN A PACKED BEDLIMITED REACTIONS IN A PACKED BED

• IN MASS TRANSFER-DOMINATED REACTIONS, THE SURFACE REACTION IS SO RAPID THAT THE RATE OF TRANSFER OF REACTANT FROM THE BULK GAS OR LIQUID PHASE TO THE SURFACE LIMITS THE OVERALL RATE OF REACTION.

• A NUMBER OF INDUSTRIAL REACTIONS ARE POTENTIALLY MASS TRANSFER-LIMITED BECAUSE THEY MAY BE CARRIED OUT AT HIGH TEMPERATURES WITHOUT THE OCCURRENCE OF UNDESIRABLE SIDE REACTIONS.

• THE BASIC EQUATIONS DESCRIBING THE VARIATION OF CONVERSION WITH THE VARIOUS REACTOR DESIGN PARAMETERS (CATALYST MASS, FLOW CONDITIONS) WILL NOW BE DEVELOPED.



• BASIC REACTION:

• MOLE BALANCE:

WHICH YIELDS FOR ,

DadC

acB

abA +→+

RATE IN – RATE OUT + GENERATION = 0

0)()()( '' =∆+∆+− zAarzzFzF ccAAA

0→∆z

0)(1 '' =+− cAA

c

ardz

zdFA



• :RATE OF GENERATION OF A PER UNIT CATALYTIC SURFACE AREA (mole/sm2)

• :EXTERNAL SURFACE AREA OF CATALYST PER VOLUME OF CATALYTIC BED (m2/m3);FOR PACKED BEDS,

• :POROSITY OF THE BED• :PARTICLE DIAMETER (m)• :CROSS-SECTIONAL AREA OF TUBE CONTAINING

THE CATALYST (m2)

''Ar

ca

pc da /)1(6 φ−=

pdφ

cA



• THE MOLAR FLOW RATE IN THE AXIAL DIRECTION IS

• IN ALMOST ALL SITUATIONS INVOLVING FLOW IN PACKED-BED REACTORS, THE AMOUNT OF MATERIAL TRANSPORTED BY DIFFUSION IN THE AXIAL DIRECTION IS NEGLIGIBLE COMPARED TO THAT TRANSPORTED BY CONVECTION. THUS,

• IS THE SUPERFICIAL MOLAR AVERAGE VELOCITY THROUGH THE BED (m/s). FOR A CONSTANT THE MOLE BALANCE EQUATION BECOMES,

AcA WAF =

cAAcA AUCWAF ==

UU

0'' =+− cAA ar

dzdCU



• AT STEADY STATE, THE MOLAR FLUX OF A TO THE PARTICLE SURFACE IS EQUAL TO THE RATE OF DISAPPEARANCE OF A ON THE SURFACE:

''AA rW −=

THE BOUNDARY CONDITION AT THE EXTERNAL SURFACE:

)(''ASAcAA CCkWr −==−

SUBSTITUTING FOR IN THE MOLE BALANCE YIELDS

''Ar

0)( =−−− ASAccA CCak

dzdCU


MASS TRANSFERMASS TRANSFER--LIMITED REACTIONS IN A PACKED BEDLIMITED REACTIONS IN A PACKED BED• IN MOST MASS TRANSFER-LIMITED REACTIONS, THE

SURFACE CONCENTRATION IS NEGLIGIBLE WITH RESPECT TO THE BULK CONCENTRATION ;THUS AND INTEGRATION WITH

THE LIMIT, , YIELDS

ASA CC >>

0=−− AccA Cak

dzdCU

0,0 AA CCz == ⎟⎠⎞

⎜⎝⎛−= z

Uak

CC cc

A

A exp0



• THE REACTOR LENGTH L NECESSARY TO ACHIEVE A CONVERSION X IS

OR

• CONVERSION IS A FUNCTION OF THE MASS TRANSFER COEFFICIENT. LET US ESTIMATE THIS COEFFICIENT FOR THE FLOW THROUGH A PACKED BED.

0

0

A

ALA

CCCX −

=

LUak

Xcc=⎟

⎠⎞

⎜⎝⎛−11ln



• CORRELATION OF THOENES AND KRAMERS FOR THE FLOW THROUGH A PACKED BED:

WHERE AND

• CORRELATION VALID FOR

31

21'' )(Re ScSh =

31

21

)1(1)

1( ⎟

⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−

=⎟⎠

⎞⎜⎝

⎛− AB

p

AB

pc

DUd

Ddk

ρµ

γφµρ

γφφ

γφ )1(ReRe'

−=

γφφ)1(

'

−=

ShSh

40001;4000Re40;5.025.0 ' <<<<<< ScφM. Lacroix External Diffusion Effects 24


• :PARTICLE DIAMETER (m)• :VOID FRACTION OF PACKED BED• :SHAPE FACTOR = (SURFACE AREA)/• :SUPERFICIAL GAS VELOCITY THROUGH BED (m/s)• :VISCOSITY (kg/sm)• :FLUID DENSITY (kg/m3)• :KINEMATIC VISCOSITY (m2/s)• :GAS-PHASE DIFFUSIVITY (m2/s)

31

)__(/6 pelletofvolumeπ=pd

2pdπ

φγ

Uµρυ

ABD


MASS TRANSFERMASS TRANSFER--LIMITED REACTIONS IN A PACKED BEDLIMITED REACTIONS IN A PACKED BEDCONCLUSION

• FOR CONSTANT FLUID PROPERTIES AND DIAMETER,

• FOR A FIXED CONCENTRATION CA THE RATE OF REACTION VARIES AS

• IF THE GAS VELOCITY IS CONTINUALLY INCREASED, A POINT IS REACHED WHERE THE REACTION BECOMES RATE-LIMITED AND CONSEQUENTLY, IS INDEPENDENT OF THE SUPERFICIAL GAS VELOCITY.

21

Ukc ∝

21'' UCkr AcA ∝∝−


MASS TRANSFERMASS TRANSFER--LIMITED REACTIONS IN A PACKED BEDLIMITED REACTIONS IN A PACKED BED• ANOTHER CORRELATION FOR THE MASS TRANSFER

COEFFICIENT (COLBURN J FACTOR):

WHERE

• CORRELATION VALID FOR BOTH GASES ( ) AND LIQUIDS ( ) IN FIXED OR FLUIDIZED BEDS

• FOR NONSPHERICAL PELLETS, THE EQUIVALENT DIAMETER IS USED: WHERE IS THE EXTERNAL SURFACE AREA OF THE PELLET.

386.082.0 Re365.0

Re765.0

+=DJφ

Re31

Sc

ShJD =

10Re >01.0Re >

π/pp Ad = pA


EXAMPLE No. 2:EXAMPLE No. 2:MANEUVRING A SPACE SATELLITEMANEUVRING A SPACE SATELLITE

• HYDRAZINE HAS BEEN STUDIED EXTENSIVELY FOR USE IN MONOPROPELLANT THRUSTERS FOR SPACE FLIGHTS OF LONG DURATION. THRUSTERS ARE USED FOR ALTITUDE CONTROL OF COMMUNICATION SATELLITES. HERE THE DECOMPOSITION OF HYDRAZINE OVER A PACKED BED OF ALUMINA-SUPPORTED IRIDIUM CATALYST IS OF INTEREST. IN A PROPOSED STUDY, A 2% HYDRAZINE IN 98% HELIUM MIXTURE IS TO BE PASSED OVER A PACKED BED OF CYLINDRICAL PARTICLES 0.25 CM IN DIAMETER AND 0.5 CM IN LENGTH AT A GAS PHASE VELOCITY OF 15 m/s AND A TEMPERATURE OF 750 K. THE KINEMATIC VISCOSITY OF HELIUM AT THIS TEMPERATURE IS 4.5 x 10-4 m2/s. THE HYDRAZINE DECOMPOSITION REACTION IS BELIEVED TO BE EXTERNALLY MASS TRANSFER-LIMITED UNDER THESE CONDITIONS. IF THE PACKED BED IS 0.05 m IN LENGTH, WHAT CONVERSION CAN BE EXPECTED? ASSUME ISOTHERMAL OPERATION. DAB=0.69 x 10-4 m2/s AT 298 K; BED POROSITY=30%.


THE SHRINKING CORE MODELTHE SHRINKING CORE MODEL

• THE SHRINKING CORE MODEL IS USED TO DESCRIBE SITUATIONS IN WHICH SOLID PARTICLES ARE BEING CONSUMED EITHER BY DISSOLUTION OR REACTION AND, AS A RESULT, THE AMOUNT OF MATERIAL BEING CONSUMED IS SHRINKING.

• EXAMPLES OF APPLICATIONS: PHARMACOKINETICS, FORMATION OF AN ASH LAYER AROUND BURNING PARTICLES AND CATALYST REGENERATION.


THE SHRINKING CORE MODEL: THE SHRINKING CORE MODEL: CATALYST REGENERATIONCATALYST REGENERATION

REMOVAL OF CARBON FROM CATALYSTS PARTICLES THAT HAVE BEEN DEACTIVATED BY FOULING.

CARBON IS FIRST REMOVED FROM THE OUTER EDGE OF THE PELLET AND THEN IN THE FINAL STAGES OF THE REGENERATION FROM THE CENTER CORE OF THE PELLET.

AS THE CARBON CONTINUES TO BE REMOVED FROM THE POROUS CATALYST PELLET, THE REACTANT GAS MUST DIFFUSE FARTHER INTO THE MATERIAL AS THE REACTION PROCEEDS TO REACH THE UNREACTED SOLID PHASE. THE REGENERATION TIME (3 HOURS) CAN BE REDUCED BY INCREASING THE GAS-PHASE OXYGEN CONCENTRATION AND TEMPERATURE.


CATALYST REGENERATIONCATALYST REGENERATIONREMOVAL OF CARBON FROM A CATALYST PARTICLE

A CORE OF UNREACTED CARBON IS CONTAINED BETWEEN r=0 AND r=R. CARBON HAS BEEN REMOVED FROM THE POROUS MATRIX BETWEEN r=R AND r=R0 . OXYGEN DIFFUSES FROM THE OUTER RADIUS r=R0 TO THE RADIUS R, WHERE IT REACTS WITH CARBON TO FORM CARBON DIOXYDE, WHICH THEN DIFFUSES OUT OF THE POROUS MATRIX. THE REACTION IS RAPID SO THE RATE OF OXYGEN DIFFUSION TO THE SURFACE CONTROLS THE RATE OF CARBON REMOVAL.


CATALYST REGENERATIONCATALYST REGENERATION

• ALTHOUGH THE CORE OF CARBON IS SHRINKING WITH TIME, WE ASSUME THE CONCENTRATION PROFILES AT ANY INSTANT IN TIME TO BE THE STEADY-STATE PROFILES OVER THE DISTANCE (R0-R).

• TO STUDY HOW THE RADIUS OF UNREACTED CARBON CHANGES WITH TIME WE FIRST FIND THE RATE OF DIFFUSION OF OXYGEN TO THE CARBON SURFACE. NEXT, WE PERFORM A MOLE BALANCE ON THE ELEMENTAL CARBON AND EQUATE THE RATE OF CONSUMPTION OF CARBON TO THE RATE OF DIFFUSION OF OXYGEN TO THE GAS CARBON INTERFACE.

• OXYGEN REACTS ONLY WHEN IT REACHES THE SOLID CARBON INTERFACE LOCATED AT r=R.

• WE SHALL LET SPECIES A REPRESENT O2.


REMOVAL OF CARBON FROM CATALYST: OREMOVAL OF CARBON FROM CATALYST: O22

• MOLE BALANCE ON O2 (A) BETWEEN AND YIELDS:

• FOR EVERY MOLE OF OXYGEN THAT DIFFUSES INTO THE SPHERICAL PELLET, ONE MOLE OF CARBON DIOXIDE DIFFUSES OUT:

• COMBINING BOTH EQUATIONS YIELDS

r rr ∆+

0)( 2

=dr

rWd Ar

drdCDW A

eAr −=

02 =⎟⎠⎞

⎜⎝⎛

drdCr

drd A

De IS THE EFFECTIVE DIFFUSIVITY OF THE CATALYST

;0),(;, 00

====

A

AA

CtRrCCRr


REMOVAL OF CARBON FROM CATALYST: OREMOVAL OF CARBON FROM CATALYST: O22

• THE SOLUTION TO THIS DIFFERENTIAL EQUATION IS

• THE MOLAR FLUX OF O2 TO THE GAS-CARBON INTERFACE IS

00 /1/1/1/1RRrR

CC

A

A

−−

=

20

0

)/1/1( rRRCD

drdCDW AeA

eAr −−

=−=


REMOVAL OF CARBON FROM CATALYST: CREMOVAL OF CARBON FROM CATALYST: C

• MOLE BALANCE ON SHRINKING CORE OF CARBON YIELDS:

• THE RATE OF DISAPPEARANCE OF CARBON = FLUX OF O2 TO THE GAS-CARBON INTERFACE:

• THE BALANCE EQUATION ON THE CARBON CORE BECOMES

CC

CrdtdR

ρφ

''

=DENSITY OF CARBON

VOLUME FRACTION OF CARBON

02

0''

/)(

RRRCDWr Ae

RrArC −=−=− =

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=−0

20

/1

RRRCD

dtdR

CC

Ae

ρφM. Lacroix External Diffusion Effects 35

REMOVAL OF CARBON FROM CATALYST: CREMOVAL OF CARBON FROM CATALYST: C

• THE SOLUTION WITH THE LIMIT AT YIELDS THE TIME NECESSARY FOR THE CARBON INTERFACE TO RECEEDE INWARD TO A RADIUS R:

0RR = 0=t

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

3

0

2

00

20 231

6 RR

RR

CDRt

Ae

CC φρ

AS THE REACTION PROCEEDS, THE REACTING GAS-SOLID MOVES CLOSER TO THE CENTER OF THE CORE.

THE TIME NECESSARY TO CONSUME ALL THE CARBON IN THE CATALYST PELLET IS:

0

20

6 Ae

CCc CD

Rt φρ=


EXTERNAL DIFFUSION EFFECTS (7) Marcel Lacroix Université ...

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