Heat and Mass Transfer Resistances.ppt

8/12/2019 Heat and Mass Transfer Resistances.ppt

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Ch E 542 - Intermediate Reactor Analysis & Design

Heat and Mass

Transfer Resistances
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Mass Transfer & Reaction

When convection dominates, the boundary

condition expressing steady state flux continuity

at z=is used;

kcis the convection mass transfer coefficient

AsAcAs CCk W


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for flow around a sphere (roughly the geometric

shape of a catalyst particle), the convective heat

transfer coefficientcan be found from correlation

such as the following:

t

p

kdhNu

pdvRe

t

Pr

3121 PrRe6.02Nu

AsAcAs CCk W


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By the heat/mass transfer analogy:

for flow around a sphere, the convective heat transfercoefficient can be found from:

PrSc

NuSh

ABt

c

Dk

kh

AB

pc

DdkSh

pdvRe

ABDSc

3121 ScRe6.02Sh


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"rW AsAs AsrCk

molar flux to catalyst surface = reaction rate on surface

AsrAsAc CkCCk

AsAc CCk

rc

AcAs

kkCkC

rc

ArcAs

kkCkk"r


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Fast Reaction Kineticsfast reaction kinetics

Ac

r

Arc

rc

ArcAs Ck

k

Ckk

kk

Ckk"r

cr kk

3121

p

ABc ScRe

dD6.0k

21

p

21

61

32

ABc

d

vD6.0k

31

AB

21

p

p

ABc

D

vd

d

D6.0k

Frssling Correlation


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Fast Reaction Kineticsfast reaction kinetics

AcAs Ck"r

cr kk

21

p

21

61

32

ABc

d

vD6.0k

TfD

61

32

AB

DAB

gas

liquid

Tas

Tfd

v21

p

21

to increase kc

v

dp


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kcvdp vdp0.5

: rAs vdp kr kcvdp CAkr kcvdp

:

0 5 10 150

0.02

0.04

0.06

0.08

0.1

rAs vdp

vdp0.5

Reaction and Mass Transfer

reaction

rate limited

mass

transferlimited


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Rate Units for Catalytic Reaction

for single pellets

for packed beds

acsurface area / gramcAA a"r'r

pcc d

6a

ppB

cd

16

d

6

a


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Example Calculation

The irreversible gas-phase reaction AB is carried out in a PBR. The

reaction is first order in A on the surface.

The feed consists of 50%(mol) A (1.0 M) and 50%(mol) inerts and enters

the bed at a temperature of 300K. The entering volumetric flow rate is

10 dm3/s.

The relationship between the Sherwood Number and the ReynoldsNumber for this geometry is

Sh = 100 Re

Neglecting pressure drop, calculate catalyst weight necessary to achieve

60% conversion of A for

isothermal operation

adiabatic operation


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Example Calculation

'rdW

dXF AAo

Mole Balance

Rate Law

AsrAs C'k'r

assume reaction is mass transfer limited

AsAcA CCkW

AAs Wr

'kk

C'kk'r

rc

ArcAs

Re100Sh

21

p

AB

pc vd100

D

dk

gs

cm4242

vd

d

D100k

321

p

p

ABc

Mass Transfer Coefficient


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Example Calculation

'rdW

dXF AAo

Mole Balance

Rate Law

gas-phase, = 0, T = T0, P = P0.Stoichiometry

X1CC AoA

'kk

C'kk'r

rc

ArcAs

gs

cm

4242k

3

c

Energy Balance

Reaction is being carried

out isothermally. Thus,

energy balance not needed

and krf(T)


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Example Calculationgas-phase, = 0, P = P0.Stoichiometry

T

TX1CC oAoA

'kk

C'kk'r

rc

ArcAs

gs

cm4242k

3

c

T1

T

1

R

E

ror

Re'kT'k

'rdW

dXF AAo

Mole Balance

Rate Law Energy Balance

io

piio

r TCF

HXT


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Multicomponent Diffusion

Exact form of the flux equation for multicomponent

mass transport:

A simplified form uses a mean effective binary

diffusivity,

1N,,2,1j,NyyDCN

N

1kkj

1N

1kkjktj

N

1k

kjjjmtj NyyDCN


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The Stefan-Maxwell equations (Bird, Stewart, Lightfoot)

are given for ideal gases:

For binary system:

N

jk 1k

kjjk

jk

jt NyNyD

1

yC

2111121t NNyND

1

yC


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Solved for flux

Simplified for

assumed equimolarcounter-diffusion

2111121t NNyND

1

yC

211112t1 NNyyDCN

112t1 yDCN


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The effective binary diffusivity for species j can then be

defined by equating the driving force terms of the

expression containing Djmand the Stefan-Maxwell

N

jk 1k

kjjk

jk

jt NyNyD

1

yC

N

1k

kjjmjtj NyDyCN


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The effective binary diffusivity for species j can then be

defined by equating the driving force terms of the

expression containing Djmand the Stefan-Maxwell

N

1k

kj

N

jk

1k

kjjk

jk

jmj NyNyNyD

1DN


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use for diffusion of species 1 through stagnant 2, 3, (all

flux ratios are zero for k=2,3,) reduces to

the "Wilke equation"

N

1k

kjj

N

jk 1k

kjjkjk

jm NyN

NyNyD

1

D

1

N

3,2k k1

k

1m1 Dy

y11

D1


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For reacting systems where steady-state flux ratios are

determined by reaction stoichiometry,

N

jk j

kjk

jkjjN

1k k

k

j

N

jk j

kjk

jk

jm

yyD

1

y1

1

y1

yyD1

D

1

constantN

j

j


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Diffusion/Rxn in Porous Catalysts

Effective Diffusivity (De) is a measure of

diffusivity that accounts for the following:

Not all area normal to flux direction is available for

molecules to diffuse in a porous particle (P) Diffusion paths are tortuous ()

Pore cross-sections vary ()

Internal void fraction, s= P

~

DD PAe


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Extended Stefan-Maxwell

Solved for binary, steady-state, 1D diffusion

Kj,e

D

jN

1k

D

kj

D

jk

jk,e

jD

NNyNy

D

1p

RT

1

KA,eAB,e0AAB

KA,eAB,eAAB

BA

AB,et

ADDyNN11

DDLyNN11

lnNN1L

DC

N


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Define effective binary diffusivity for use in

single reaction multicomponent systems:

dz

dCDN j

jm,ej

Kj,e

N

1k j

kjk

jk,ejm,e D

1yyD

1

D

1


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Quantify De

Random Pore Model

Parallel Cross-linked Pore Model

Pore Network Model of Beeckman & Froment

Tortuosity factor using Wicke-Kallenbach cell

Pore diffusion with

Adsorption Surface Diffusion


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steady state mass balance

rate in at r

r

2

Ar r4W rate out at r + r

rr

2

Ar r4W

rate of generation within shell

cmasscatalyst

ratereaction

volumeshell

masscatalyst volumeshell

rr + r

R

rr4

2

m'

Ar

0rr4r

r4Wr4W

2

mC

'

A

rr

2

Arr

2

Ar

0rr

dr

rWd 2C

'

A

2

Ar BA cat


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0rSCkrdr

dC

Ddr

d 2Ca

n

An

2A

e

identify boundary conditions

finiteC0rA

symmetry

AsRrA CC

surface

dimensionless

As

A

C

C

R

r

AsA

Cd

dC

R

1

d

dr

R

C

d

d

dr

dC AsA

0CD

Sk

dr

dC

r

2

dr

Cd nA

e

CanA

2

A

2

0D

CRSk

d

d2

d

d n

e

1n

As

2

Can

2

2


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define Thiele modulus (n)

0D

CRSk

d

d2

d

d n

e

1n

As

2

Can2

2

e

1n

As

2

Can2

nD

CRSk

0dd2

dd n2

n2

2

understand the Thiele modulus

R0CDRCSk

Ase

nAsCan2

n reaction rate

di f fusion rate

large n- diffusion controlssmall n- kinetics control


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Diffusion/Rxn in Porous Catalystsfirst order

kinetics

(n = 1)

define y =

0d

d2

d

d 212

2

2

e

Can2

1 RD

Sk

322

2

2

2 y2

d

dy2

d

yd1

d

d

2

y

d

dy1

d

d

0yd

yd 212

2

1111 sinhBcoshAy

1B

1

Asinhcosh 11

differential has the solution apply boundary conditions

1,1

finiteis,0


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Diffusion/Rxn in Porous Catalystsfirst order

kinetics

(n = 1)

0d

d2

d

d 212

2

2

e

Can2

1 RD

Sk

0yd

yd 212

2

1111 sinhBcoshAy

1B

1

Asinhcosh 11

differential has the solution apply boundary conditions

1,1

finiteis,0

1

1

sinh

sinh1

As

A

C

C


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Internal Effectiveness Factor ()

The internal effectiveness factor () is a measure

of the relative importance of diffusion to reaction

limitations:

sAsT,Ctoexposedweresurfaceentireifrate

ratereactionoverallactual

As

A"

As

"

A'

As

'

A

As

A

MM

rr

rr

rr

M mol / timer mol / time / mass cat


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Determine MAs(rate if all surface at CAs)

catalystmasscatalystmass

areasurfaceareaunitperrateMAs

'

Asr

aS

CVAsM

x

x

As1Ck

c

3

34

aAs1As RSCkM


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Determine MA(actual rate is equal to reactant

diffusion rate at outer surface)

1

AseA

d

dCRD4M

11

12

1

11

1 sinhsinh1

sinhcosh

dd

1coth 11

1cothCRD4M 11AseA


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Substitute results into definition of

As

A

M

M

c

3

34

aAs1

11Ase

RSCk

1cothCRD4

1cothRSk

D3 11

c

2

a1

e

1coth3 1121


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Revisit and

Thiele modulus -

Derived for spherical particle geometry

Derived for 1storder kinetics

For large , approximately

Internal effectiveness factor -

Assumed =0, correction applied when 0

Assumed isothermal conditions

2

1

21

31n

2


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Non-Isothermal Behavior

For exothermic reactions, can be > 1 as internaltemperature can exceed Ts.

The rate internally is thus larger than at the surfaceconditions where is evaluated.

The magnitude of this effect is dependent on Hrxn, Ts, Tmax, and kt(thermal conductivity of the pellet)

and are used to quantify this effect:

can result in mulitple steady states

No multiple steady states exist if Luss criterion is fulfilled

NumberArrheniussRT

E

st

Aserxn

s

smax

Tk

CDH

T

TT

14


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Overall Effectiveness Factor

When both internal AND external diffusion

resistances are important (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

bulk conditions, rather than surface conditions:

bulkA,Ctoexposedweresurfaceentireifrateratereactionoverallactual


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Accounting for reaction both on and within the pellet, the molar ratebecomes:

For most catalyst, internal surface area is significantly higher than the

external surface area:

V1SarM cac"

AA

b

bac"

AcA SaraW

ba

"

AcA SraW


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ba

"

AcA SraW reaction rate(internal & external surfaces)

VaCCkVaW cAsbulk,AccAr

mass transport rate

internal surfaces not

all exposed to CAs

As1

"

As

"

A Ckrr Relation between CAsand CAdefined by the as:

VSCkVaW baAs1cA

baAs1cAsbulk,Ac SCkaCCk


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ba

"



mass transport rate

As1

"

As

"

A Ckrr Relation between CAsand CAdefined by the as:

ba1cc

bulk,Acc

As Skka

Cka

C Solving for CAs:


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ba

"



mass transport rate

ba1cc

bulk,Acc1"

ASkka

Cakkr

Substitution into the rate law:

ba1cc

bulk,Acc

As Skka

Cka

C Solving for C

As:


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Overall Effectiveness Factorsummary of factor relationships:

ba1cc

bulk,Acc1"

ASkka

Cakkr

Rearranging the expression:

bulk,A1

ccba1

CkakSk1

"

bulk,A

"

A rr ccba1 akSk1

"As

"bulk,A

"A rrr

As1

"

As

Ckr

Ab1

"

Ab Ckr

Overall Effectiveness Factor ()


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Weisz-Prater Criterion

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

CWPis the known as the Weisz-Prater parameter. All

quantities are known or measured.

CWP> 1, severe diffusion limitations

Ase

c

2'

obs,A2

1WPCD

RrC


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Mears Criterion

Mass transfer effects negligible when it is true that

n is the reaction order, and the transfer coefficients kcand h (below) canbe estimated from an appropriate correlation (i.e., Thoenes-Kramers for

packed bed flow)

Heat transfer effects negligible when it is true that

15.0Ck

nRr

Abc

b

'

A

15.0ThR

RErH2

bg

b

'

Arxn


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A li i PBR


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Application to PBRs

Which can be rewritten as:

AbabAb C

UkS

dzdC

Entrance condition:oAb0zAb

CC

Integrating and applying boundary condtion yields:

U

zkS

expCC

ab

AbAb o

Heat and Mass Transfer Resistances.ppt

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