Internal Diffusion

8/6/2019 Internal Diffusion

1/45

Chemical Reaction Engineering

Lecturer :

Lecture 12


2/45

This course focuses on internal diffusion effects on

heterogeneous reactions.

Internal diffusion: diffusion of the reactants or products from the external

pellet surface (pore mouth) to the interior of the pellet. (Chapter 12)

When 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

the entire catalytic surface is not accessible to the same concentration.

CAbCAs

C(r)


3/45

Diffusion and reaction in a porous

spherical catalyst pellets

The pores are a series of tortuous, interconnecting paths

of pore bodies and pore throats with varying cross-

sectional areas.

An effective diffusion coefficientis used to describe the average diffusion

taking place at any position rin the pellet.The radial flux is based on the

total area normal to diffusion transport.The effective diffusivity:

X

WJ

~p

Ae DD !

AD

pJ

W

X~

the bulk diffusivify

tortuosity (typical ~ 3.0)

pellet poro sity (typical ~0.4)

constriction factor (typical ~0.8)


4/45

An irreversible isomerization reaction take places on the surface of the pore walls

within the spherical pellet of radius R :

BA p

rR

CAs

Rate of A in at r= WAr area = rAr rW24Tv

Rate of A out at r - (r= WAr area = rrAr rW (v24T

The mole balance over the shell thickness(ris:

0444 222 !v(vdvv( cmArrArrAr rrrrWrW VTTT

whereVc

is the mass of catalyst per unit volume;

rm is the mean radius between rand r- (r

0

22

!vvd cAAr rr

dr

rWdV


5/45

02

2

!vvd cAAr rr

dr

rWdV

For EMCD, or dilute concentration

!

dr

dCDW AeAr

02

2

!vvd

cA

Ae

rrdr

rdr

dd

V

aAA Srr dd!d where is the rate of reaction per unit surface area;Sa is the surface area of the catalyst per unit mass of catalyst

typical value ofSa is150 m2/gof catalyst

Ardd

nAnA Ckr !dd nth ordersurface reaction

02

2

!

a

n

Anc

Ae

SCkrdr

rdr

dCDd

V0

22

2

!

nA

e

ancAA CD

Sk

dr

dC

rdr

Cd V


6/45

02

2

2

!

nAe

ancAA CD

Sk

dr

dC

rdr

Cd V B.C.CA = CAC = constant at r=0

CA = Cas at r= R

As

A

C

C!N

R

r!P

02 2

2

2

!

nn

d

d

d

dNJ

NN

e

n

Asancn

D

CRSk12

2

!V

J

B.C.

N = finite value at P =0

N = 1 at P = 1

Dimensionless form of equations

describing diffusion and reaction

The Thiele modulus, Jn

ratediffusiona

ratereactionsurfacea

R

CD

RCSk

D

CRSk

Ase

n

Asanc

e

n

Asancn

!

!!

)0(

122 VVJ

Jn o internal diffusion limits the overall rate of reaction

Jn

q surface reaction limits the overall rate of reaction


7/45

For a reaction : BAp

If the surface reaction is rate-limiting with respect to the adsorption of A and desorption of

B, and ifspecies A and B are weakly adsorbed (low coverage) and present in diluteconcentration, the apparent first-order rate law:

AA Ckr 1!dd

02 2

2

2

!

nn

d

d

d

dNJ

NN B.C.N = finite value at P =0

N = 1 at P = 1

02 2

12

2

!

NJP

N

PP

N

d

d

d

d

e

ac

D

SkR

11

VJ !

PJPPJP 1111 sinhcosh

BA!

N = finite value at P =0

N = 1 at P = 1

!

1

1

sinh

sinh1

J

PJ

PN

N

As

A

C

C

!N

R r=0

smallJ1

medium J1large J1

small J1: surface reaction control and a significant amount of reactant diffuses well into the pellet

interior without reacting;

large J1: surface reaction is rapid and the reactant is consumed very closed to the external pellet surface

(A waste of precious metal)


8/45

Internal effectiveness factor:(1) the relative importance of diffusion and reaction limitations

(2) a measurement ofhow far the reactant diffusesinto the pellet before reacting

conditionssurfaceexternalthetoedexposweresurfacerinterioentireifreactionofrate

reactionofrateoverallactual!L

As

A

rr

d!L observed reaction rate

As

A

As

A

M

M

catalystofmassr

catalystofmassr!

v

vd!

)(

)(L

)()( timecatalystofmass

mole

v

time

mole

vd!

vv! cAscaAsAs RrRSCkM VTVT33

13

4

3

4)(

first-order reaction surface area per unit mass of catalyst

(Cas)


9/45

1

2 444 !!! !

!

! N

TTTd

dCRD

R

rd

CCd

CRDdr

dCDRM AseRr

As

A

AseRrA

eA

The actual rate of reaction:

(the reaction at which the reactant diffusesinto the pellet at the outersurface at the S.S.)

1coth4 11 ! JJT AseA CRDM

e

ac

D

SkR 11

VJ !

1coth3

1coth

34

4

34

1coth4

112

1

11

31

3

11

!

!

vd

!!

JJJ

JJ

VT

T

VT

JJTL

caAs

Ase

cAs

Ase

As

A

RSCk

CRD

Rr

CRD

M

M

pp750 forJ1 vs.L


10/45

1coth3

112

1

! JJJ

Le

ac

D

SkR

11

VJ !

11 pqq LJR surface-reaction-limited

1,33

)30(,11

1 !$}""ac

e

Sk

D

R VJLJ

diffusion-limited reaction (external

diffusion will have a negligible effect on the

overall reaction rate)

Overall rate of reaction for a first-order reaction:

1coth3

112

1

!d

d! JJ

JL

As

A

r

rArd aAsAsA SCkrr 1LL !d!d

ac

e

Sk

D

R VL

1

3$ internal-diffusion-limited

As

c

aeA C

kSD

Rr

V13!d

How to increase the rate of reaction?

(1) decrease the radius R

(2)increase the temperature

(3)increase the concentration

(4)increase the internal surface area


11/45

For reaction of ordern, the Thiele modulus, Jn

e

n

Asancn

D

CRSk12

2

!V

J

When the reaction isinternal-diffusion-limited ( ) :""nJ

2/12/12/1

3

1

23

1

2 nAs

anc

e

n

CSk

D

Rnn

!

!VJ

L

When reaction order n isgreater than 1:The effectiveness factor decreases withincreasing concentration at the external pellet surface.

Application: use the effectiveness factor to calculate the true reaction order

Disguised/Falsfied Kinetics (the internal-diffusion-limited reaction)


12/45

For a reactionn

AsnA Ckrdd!d

ln CAs

ln -rA

slope = napparent reaction order

As

A

r

r

d!L nAsanAsA CSkrr LL !d!d

When the reaction isinternal-diffusion-limited ( ) :

nn JL

3

1

22/1

!

""n

J

2/1

2

12/1

)1(

233

1

2

!

!d

n

Asn

c

aen

Asan

n

A Ckn

SD

RCSknr VJ

From experimental results, we have:

(true reaction rate)

(measured reaction rate)

nAsn

n

Asn

c

aeA CkCk

n

SD

Rr

d d!

!d 2/121

)1(

23

V 21 n

n

!dThe relation between the apparent

reaction order and the true

reaction order


13/45

Apparent activation energy Vs. True activation energy

For a reaction, we obtain the experimental results and the apparent activation energy:

n

AsnA Ckrdd!d (measured reaction rate constant)

RT

E

appn

app

eAk

!d

nAsn

n

Asn

c

aeA CkCk

n

SD

Rr

d d!

!d 2/121

)1(

23

V

true rate of reaction apparent rate of reaction

RT

E

truen

true

eAk

! RTE

appn

app

eAk

!d

RT

ECA

RT

ECA

n

SD

R

appn

Asapptruen

Astrue

c

ae !

dln

2)1(

23ln

2/12

1

V

apptrue EE 2! The relation between the apparent activation energy and the true activation energy


14/45

The importance of the internal-diffusion-limited reactions:

Disguised/Falsfied Kinetics (the internal-diffusion-limited reaction)

When a reaction takes place and the rate of reaction is developed:

n

AsnA Ckrdd!d

trueapp EE2

1!

2

1 n

n

!d

apparent activation energy

apparent reaction order

If the pellet size became smaller and the reaction is no more internal-diffusion-limited

Wrong reaction order and activation energy might be used for the design of reactor!

Runaway reaction conditions might occur and the exploding situation might happen!


15/45

Quick estimation of the rate-limiting step in a heterogeneous reaction

1coth3)( 1121

!d

d! JJJ

LAs

A

r

obsr

e

ac

D

SkR 11

VJ !

The internal effectiveness factor:

1coth3 112

1 ! JJLJ

the Weisz-Prater parameter

Ase

cA

Ase

aAsc

As

A

e

ac

As

A

As

As

WP

CD

Robsr

CD

SCkR

r

obsr

D

SkR

r

obsrratediffusiona

reactionofrateactualobserved

ratediffusiona

Catevaluatedreactionofrate

Catevaluatedreactionofrate

reactionofrateactualobservedC

2

1212

)(

)()(

)(

)(

V

VV

d!

vd

d!v

d

d!

!

v!

When CWP >>1; internal diffusion limited

When CWP


16/45

ExampleA first-order reaction Ap B was carried out over two different-sized pellets.If the external

mass transfer resistance is negligible, estimate the Thiele modulus and effectiveness factor

for each pellet according to the following experimental results. What should the size of the

pellet to eliminate the internal diffusion resistance?

easured rate of reaction

(mol/g cats)v 105

Pellet radius

(m)

Run 1 3 0.01

Run 2 15 0.001

1coth3)(

11

2

1

2

!!d

! JJLJV

Ase

cAW

CD

RobsrC

1coth

1coth

)(

)(

1212

1111

2

22

2

11

!

d

d

JJ

JJ

Robsr

Robsr

A

A

e

ac

D

SkR 11

VJ ! 10

2

1

12

11

!! RR

JJ

1coth

1)10coth(10

001.015

01.03

1212

1212

2

2

!vv

JJJJ

65.1

5.16

12

11

!

!

J

J


17/45

1coth3 112

1 ! JJLJ

2

1

11 1coth3

J

JJL

!

65.1

5.16

12

11

!

!

J

J 856.0

182.0

2

1

!

!

L

L

Eliminating the internal diffuse resistance p L } 1Assuming L =0.95

2

13

1313 1coth3

95.0 J

JJ

! 9.013 !J

3

1

13

11

R

R!

JJ

mR

4

3 105.5

v!


18/45

When both internal and external diffusion are important

CAbC

As

C(r)

At steady-state:the transport of the reactants from the bulk fluid to the

external surface of the catalyst is equal to the net rate of

reaction of the reactant within and on the pellet

The molar rate of mass transfer from the bulk fluid to the

external surface:

VaWM cArA (!

molar flux

external surface area per unit reactor volume

reactor volume

This molar rate of mass transfer to the surface is equal to the net rate of reaction on and

within the pellet:

arearnalinteareaexternalrM AA dd!


19/45

arearnalinteareaexternalrM AA dd!

Vac( VS ba (V

catalystofmass

areaalintern

volumereactor

catalystofmassvolumereactor

VSVarM bacAA ((dd! V

VSVarVaWM bacAcArA ((dd!(! V

usually small compared with the next term

baAscAsAbc SraCCk VL dd! )(

AsAbcAr CCkW !

)( AsA rr dd!dd L

Assume a first-order reaction AsAs Ckr 1!dd baAscAsAbc SCkaCCk VL ! )( 1


20/45


21/45

Mass transfer and reaction in a packed bed reactor

z=0 z= L

Ac

z z+(z

At steady-state, the mass balance on A over the volume element (V=Ac(zis:

0!(d ( zArAWAW cbAczzAzczAz V 0!d bAAz rdz

dWV

[rate in] - [rate out] +[rate of generation] =[rate of accumulation]

AbAb

ABAz UCdz

dCDW !

BAp

02

2

!d bAAbAb

AB rdz

dCU

dz

CdD V

the overall reaction rate within and on the catalyst

per unit mass of catalyst:

;vd!d AbA rrfirst-order reaction,

aAbaAbAb SkCSrr !dd!d

aAbA SkCr ;!d

02

2

!; baAbAbAb

AB SkC

dz

dCU

dz

CdD V

U


22/45

02

2

!; baAbAbAb

AB SkCdz

dCU

dz

CdD V

If the axial dispersion is neglected, and the boundary condition at the entrance of the reactor:

00 !! zatCC AbAbU

zkS

AbAb

ab

eCC)(

0

;

!V

Example

A 2% NO-98% air mixture flows at a rate of 1 x 10-6 m3/s through a 2-in-ID tube packed

with porous carbonaceoussolid at a temperature of 1173K and a pressure of 101.3 kPa.The

reaction is first-orderin NO, calculate the weight of poroussolid necessary to reduce the

NO concentration to a level of0.004%.

22

1NCOCNO p

NOaNO CkSr !d

998.02

004.02

0

0 !

!

!Ab

AbAb

C

CCX Our purpose:X= f(W)


23/45

NO down the length of the reactoris: (when the axial dispersion is neglected)

0!; baAbAb SkC

dz

dCU V

zAW cbV!

u

SkC

dW

dC aAbAb ;!

cUAu !

;!

0

0 expu

WkSCC aAbAb

00 !! WatCC AbAbB.C.

Dilute:0;1 uu !I

;!!

00

exp11v

WkS

C

CX a

Ab

Ab

X =f(W)XkS

vW

a ;!

1

1ln0

ccba akkS VL

L

!;1

2

1

11 1coth3

J

JJL

!

e

ac

D

SkR 11

VJ !

external transfer coefficient

31

21

eR SchS d!dv

Udp

)1(eR

J!d

ABD

vSc !

AB

pc

D

dkhS

J

J

!d

1

smkc /1065v!

167.0!L

external diffusion resistance isimportant


24/45

ccba akkS VLL

!;

1

32

3

2

/500)1(6

)1(

3

4

4mm

dr

ra

p

c !

!!J

J

T

T

059.0!;

smkc /1065v!

167.0!L

internal diffusion resistance isimportant

XkS

vW

a ;!

1

1ln0 g

gmsmm

smW 450

998.01

1ln

)059.0)(/530)(/1042.4(

/10122310

36

!v

v!

In this example, bothinternal and external diffusion resistance are significant.Which one is the rate-limiting step? See page 768Table 12-1


25/45

Multiphase reactors

Two or more phases are necessary to carry out

the reaction

Examples (refer to Table 12-2 on page 769)

Slurry reactor

Trickle bed reactor

Fluidized bed reactor


26/45

Slurry reactors

reactant gasis bubbled through a solution

containing solid catalyst particles

solution may be either a reactant or an inert

batch/continuous

temperature control and heat recovery are

relatively easy


27/45

An industrial slurry reactor used to convert synthesis gas (CO+ H2) to a

hydrocarbon wax by the Fischer-Tropschsynthesis.

OHHCHCO 252252 255125 p

Reactant gas

Recycle gas

product

catalyst

Absorption from the gas phase intothe liquid phase at the bubble surface

Diffusion in the liquid phase from the

bubble surface to the bulk liquid

Diffusion from the bulk liquid to the

external surface of the solid catalyst

Internal diffusion of the reactant in

the porous catalyst

Reaction within the porous catalyst

These steps are resistance to the overall rate of reaction.


28/45

Consider the hydrogenation of methyl linoleate to form methyl oleate in a slurry reactor:

OHL p 2

GasAbsorption The rate of absorption of H2per unit volume of linoleate oil is:

)( AbAibbA CCakR !

mass transfer coefficient

bubble surface area

H2 concentration at oil-bubble interface

H2 concentration in the bulksolution

Transport to the catalyst pellet The rate of mass transfer of H2 from the bulksolution

to the external surface of catalyst pellet is:

)( AsAbpcA CCmakR !

mass transfer coefficient

external surface area of pellets

H2 concentration at external surface of

catalyst pellet

mass concentration of catalyst


29/45

Diffusion and reaction in the catalyst pellet The rate of reaction per volume ofsolution:

)( AsA rmR d! L

internal effectiveness factor rate of reaction if the entire interior of the pellet were

exposed to the reactant concentration at the external surface

The rate law The rate law is (linoleate in the liquid phase isin excess):

AA kCr !d

At steady-state, all rates are equal:

AsAsAsAbpcAbAibbA kCmrmCCmakCCakR LL !d!!! )()()(

Ai

pcbb

A Ckmmakak

R !

L111

eliminate CAb, CAs

!

LkakmakR

C

pcbbA

Ai 1111


30/45

!

LkakmakR

C

pcbbA

Ai 1111 rcbA

Ai rrm

rR

C!

1

resistance to gas absorption

resistance to transport to catalyst surface

resistance to diffusion and reaction within the catalyst

resistance to diffusion

A

Ai

R

C

m

1

br

slope = diffusion and reaction resistance

A

Ai

R

C

m

1

gas absorption control

diffusion and reaction control

Which one is the rate limiting step?

External diffusion?

Internal diffusion?

Surface reaction?


31/45

Lkak pc

11 Combination of the diffusion and reaction resistance

IF the particles are small and therefore resistance issurface reaction control: 1}L

kkak

rrpc

rc

111}!

Lresistance isindependent of particle size

IF the particles are moderate and internal diffusion control, the internal effective factoris:

e

ac

D

kSR

VJ !1

2

1

11 1coth3

J

JJL

! large value ofJ1

1

3

JL !

ac

e

p kS

D

d VL

6!

resistance varies linearly with particle size.

IF the particles are moderate and external diffusion control:

ppc

rc d

kak

rr w!

L

11

Lkak

rrpc

rc

11!


32/45

pc

cak

r1

!cp

cp

p

cd

d

d

pelletofmass

areaexternala

VVT

T 6

6

3

2

!!!

31

21

ScRe6.02 !Sh

No shear stress between particles and fluid (diffusion to a particle in a stagnant fluid):

2!Shp

ABc

d

Dk 2! 2

1p

pc

c dak

r w!

Shear between particles and fluid

21

w

v

Ud

D

dk p

AB

pc 5.11

p

pc

c dak

r w!3121 ScRe6.0!Sh2

1

w

p

cd

Uk

Particles move with the fluid:

increasing the stirring has no effect in increasing the overall rate of reaction

Shear between the particles and fluid isimportant:

increasing the stirring increases the overall rate of reaction


33/45

ln dp

rc rr ln

s=0, reaction-limited

s=1, internal-diffusion-limited

s=1.5 ~ 2.0, external-diffusion-limited

The slope should be 0, 1, 1.5, 1.7 and 2.0.

If the slope is between these values, thissuggests that more than one resistance is limiting.

Example

A catalytic hydrogenation reaction is carried out in a slurry reactor. Hydrogen is bubbled

up through the liquid and catalyst.The experimental data are shown in the table 12-5-1 on

page 780.Determine the major resistance.

rcbA

Ai rrm

rR

C!

1A slurry reactor: rcb

H

iHrr

mr

r

C!

1

2

2


34/45

rcbH

H

H

iHrr

mr

r

H

r

C!

d!

1

2

2

2

2

Henrys law

2

2

H

iH

r

C

m

1

40Qm

80Qm

For40Qm pellet, the slope (i.e., the combination external and internal diffusion andreaction resistance)is (r

c+ r

r)=0.14; For80Qm pellet, the slope is (r

c+ r

r)=0.28

2

14.0

28.0

40

80 !!

mrc

mrc

rr

rr

Q

Q prc drr w Internal diffusion is the controllingresistance of the three resistance.

For80Qm pellet, the slope is (rc + rr)=0.28; the intercept (i.e., the gas absorptionresistance)is0.08

mr

C

H

iH 128.008.0

2

2 !

m = 0.4

08.0

08.084.0 !

b

r

r

r

rb

(rc + rr)

84.0

2

2 ! H

iH

r

C


35/45

Slurry reactor design example

A 2 m3 slurry reactoris used to convert methyl linoleate (ML) to methyl oleate (MO),

whichis the same reaction as the example in the previousslide.The molar feed rate of MLis0.7 kmol/min.The partial pressure of H2 is 6 atm and the reactoris well-mixed. Calculate

the catalyst charge necessary to achieve 30% conversion for a 60Qm particle size.

At steady state, for a slurry reactor:

From mass balance: (similar to CSTR)

A

A

r

XFV

! 0

rcbA

Ai rrm

rr

C!

1

rcbA

Ai rrm

rXF

VC!

1

0

the gas absorption resistance is0.08

214.028.0

40

80 !!

mrc

mrc

rrrr

Q

Q

21.060

!mrc

rrQ

014.02

!d!HAi PHC

3/9.3 mkgm !


36/45

Trickle bed reactor

Gas and liquid flow (trickle) concurrently down a packed

bed of catalyst particles ranging from 1/8 to 1/32 in.in

diameter The pores of the catalyst are filled with liquid

In petroleum refining, P= 34 ~ 100 atm; T= 350 ~ 425 C

Used in process:

hydrodesulfurization ofheavy oil stocks

hydrotreating of lubricating oils

other reactions


37/45

liquidgas The stepsinvolving reactantA in the gas phase are:

Transport from the bulk gas phase to the gas-liquid interface

Equilibrium at the gas-liquid interface

Transport from the interface to bulk liquid

Transport from the bulk liquid to external catalyst surface

Diffusion and reaction in the pelletSimilar to slurry reactor...

gas liquid solid

CA

CAb CAsCA(g)

Assuming a first-order reaction in dissolved gas A and in liquidB,

following similar procedures of a slurry reactor, forreactant gasA:

)(

)(

111)1()1(

1

gA

Bspcil

c

ig

cA

gAreactionAreactionA

CHkCakakaHk

r

CH

kPkr

!

d

!!d

LVJVJ

gas phase mass transfer resistance

liquid phase mass transfer resistance

external diffusion resistance

internal diffusion +rxn resistance


38/45

)(

111)1()1(gA

Bspcil

c

ig

cA C

HkCakakaHkr !

d

L

VJVJ)()( gAvggApelletointgasforoverallA CkCkr !!d

Assuming a first-order reaction in dissolved gas A and in liquid B, forreactant liquid B:

)()(

)(

11

lBvllBpelletointliquidforoverallB

lB

Aspc

B

CkCkr

CkCak

r

!!d

!

d

L

Mole balances on A and B:

)(gAvgAA Ckr

dW

dF!d!

)(lBvlBB Ckr

dWdF

!d!

These equations are solved simultaneously.


39/45

Fluidized-Bed Reactor (FBR)

The fluid velocity issufficient to suspend the particles,

but not large enough to carry them out of the vessel.

They can process large volume of fluid. They provide excellent solids mixing.

The fluidized mediumis either a liquid or gas

Above the bed is a space, termed the disengaging

section, to collect solids caught in the gasstream to fallbackinto the bed.


40/45

Bubble

Wake phase

Drift phase

The Kunii-Levenspiel bubbling bed model

As the bubbling rise, mass transfer of the

reactant gases diffuse in and out of the

bubble to contact the solid particles.

The reaction product is formed.

The product gases flow backinto a bubble. The bubble reaches the top of the bed and

the product is collected.

Determine:

The velocity at which the bubbles move

through the column.

The rate of transport of gasesin and out of

the bubbles.


41/45

From fluid mechanics:021.0029.0

3

272.0586.0

!

c

g

pg

mf

d V

V

LV

Q]I ? A

mf

mf

gc

p

mf gd

u

I

IVV

Q

]

!

1

)(

150

)(32

!

tbm

bbm

D

h

d

dd 3.0exp

4.00 )(652.0 mfcbm uuAd !

From mass transfer:

2/10 71.0 bmfb gduuu !

!

4/5

4/12/1

85.55.4bb

mf

bcd

gD

d

uK

2/1

378.6

!

b

bmf

ced

DuI

bubble-to-cloud cloud-to-emulsion

Fraction of the bubblesin the bed:)1(

0

EH

!mfb

mf

uu

uu

Volume of catalyst in the bubbles, Kb, clouds, Kc, and emulsion, Ke

! EI

IIK

)/(

)/(3)1(

mfmfb

mfmf

mfcuu

ucmfe r

!

H

HIK

1)1(

001.001.0 tob !K

particleofareasurface

volumesameofsphereofareasurface!]


42/45

The catalyst weight necessary to achieve a given conversion is:

XKk

uAW

Rcat

mfbcc

!

1

1ln

)1)(1( HIV

where

ce

cat

e

c

bc

catbR

Kk

K

kK

!

K

K

K

1

1

1

1

kcat is the specific reaction rate


43/45

Chemical vapor deposition reactors

One of the key stepsin the chip-making processis the

deposition of different semiconductors and metals on

the surface of the chip. Horizontal low-pressure CVD (LPCVD) operates at

ca. 100Pa.

process a large number of wafers without detrimental effects

to film uniformity Re < 1

As the reactant gases flow through the annulus, the reactants

diffuse from the annulus radially inward between the wafers

to coat them.


44/45

LPCVDmodelling

)(2)()(2 gsg HSiSiH m

The reactant gas flows through the annulus between the outer edges of the cylindrical

wafers and the tube wall. Silicon is to be deposited on wafers:

)(22

2)(2

2)(2

g

s

g

HSSH

SHSiSSiH

SSiHSSiH

my

ypy

ymCVD mechanisms:Rt

Mass balance on A:

02222 !(vddvvv ( rrrrlWrlW ArrArrAr TTTr

Rwl

r

dr

rWd

r

AArdd

!2)(1


45/45

annulustheinconditionsthetoedexposissurfacewaferentirewhenreactionofrate

reactionofrateoverallactual!L

)()(2

/22

)(2

)2(

)(2

)2(

2

22

101

11

2

1

222

0

JJ

J

TP]

T

T

T

T

T

TL

P

I

I

lDR

d

d

kyR

lRdr

dyD

rR

lRW

rR

drrr

ABAAw

wRrA

AB

AAw

wRrAr

AAw

R

AW

W

w

!

!

!dd

!dd

dd!

!!!

l

r

dr

rWd

r

AArdd

!2)(1

dr

dyDW A

ABAr !

AA kyr !dd0

21!

lD

ky

dr

rdy

dr

d

r AB

AA

wR

r

!AA

A

yy!]

01 2

1 !

]J

]P

PP d

d

d

d

)()( 1010 PJPJ]BKA

I

!

B.C.

)(

)(

10

10

J

PJ]

I

I

y

y

AA

A !!I0 is the modified Bessel function

P = r/RwJ1 = the Thiele modulus

lD

kR

AB

w22

!

the effectiveness factor

The value of the Thiele modulus affects the thickness of the deposition.

Internal Diffusion

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