Heat and Mass Transfer Resistances

51
Ch E 542 - Intermediate Reactor Analysis & Design Heat and Mass Transfer Resistances

Transcript of Heat and Mass Transfer Resistances

Page 1: Heat and Mass Transfer Resistances

Ch E 542 - Intermediate Reactor Analysis & Design

Heat and MassTransfer Resistances

Page 2: Heat and Mass Transfer Resistances

Mass Transfer & Reaction

• When convection dominates, the boundary condition expressing steady state flux continuity at z= is used;

– kc is the convection mass transfer coefficient

AsAcAs CCk W

Page 3: Heat and Mass Transfer Resistances

Mass Transfer & Reaction

– for flow around a sphere (roughly the geometric shape of a catalyst particle), the convective heat transfer coefficient can be found from correlation such as the following:

t

p

k

dhNu

pdvRe

t

Pr

3121 PrRe6.02Nu

AsAcAs CCk W

Page 4: Heat and Mass Transfer Resistances

Mass Transfer & Reaction• By the heat/mass transfer analogy:

– for flow around a sphere, the convective heat transfer coefficient can be found from:

PrSc

NuSh

ABt

c

Dk

kh

AB

pc

D

dkSh

pdvRe

ABDSc

3121 ScRe6.02Sh

Page 5: Heat and Mass Transfer Resistances

Mass Transfer & Reaction

"rW AsAs AsrCk

molar flux to catalyst surface = reaction rate on surface

AsrAsAc CkCCk

AsAc CCk

rc

AcAs kk

CkC

rc

ArcAs kk

Ckk"r

Page 6: Heat and Mass Transfer Resistances

Fast Reaction Kineticsfast reaction kinetics

Acr

Arc

rc

ArcAs Ck

k

Ckk

kk

Ckk"r

cr kk

3121

p

ABc ScRe

d

D6.0k

21

p

21

61

32AB

c d

vD6.0k

31

AB

21

p

p

ABc D

vd

d

D6.0k

Frössling Correlation

Page 7: Heat and Mass Transfer Resistances

Fast Reaction Kineticsfast reaction kinetics

AcAs Ck"r cr kk

21

p

21

61

32AB

c d

vD6.0k

TfD

61

32AB

DAB

gas

liquid

T as

Tfd

v21

p

21

to increase kc

v

dp

Page 8: Heat and Mass Transfer Resistances

Slow Reaction Kineticsslow reaction kinetics

Arc

Arc

rc

ArcAs Ck

k

Ckk

kk

Ckk"r

cr kk

kr is independent of• fluid velocity• particle size

Page 9: Heat and Mass Transfer Resistances

kc vdp vdp0.5 rAs vdp

k r kc vdp CA

k r kc vdp

0 5 10 150

0.02

0.04

0.06

0.08

0.1

rAs vdp

vd p0.5

Reaction and Mass Transfer

reactionrate limited

masstransferlimited

Page 10: Heat and Mass Transfer Resistances

Rate Units for Catalytic Reaction

for single pellets

for packed beds

ac surface area / gramcAA a"r'r

pcc d

6a

ppB

c d

16

d

6a

Page 11: Heat and Mass Transfer Resistances

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 Reynolds Number for this geometry is

Sh = 100 Re½

• Neglecting pressure drop, calculate catalyst weight necessary to achieve 60% conversion of A for

– isothermal operation– adiabatic operation

Page 12: Heat and Mass Transfer Resistances

Example Calculation

'rdW

dXF AAo

Mole BalanceMole Balance

Rate LawRate Law

AsrAs C'k'r

assume reaction is mass transfer limited

AsAcA CCkW

AAs Wr

'kk

C'kk'r

rc

ArcAs

½Re100Sh

21

p

AB

pc vd100

D

dk

gs

cm4242

vd

d

D100k

321

p

p

ABc

Mass Transfer CoefficientMass Transfer Coefficient

Page 13: Heat and Mass Transfer Resistances

Example Calculation

'rdW

dXF AAo

Mole BalanceMole Balance

Rate LawRate Law

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

X1CC AoA

'kk

C'kk'r

rc

ArcAs

gs

cm4242k

3

c

Energy BalanceEnergy Balance

Reaction is being carried out isothermally. Thus,

• energy balance not needed• and kr f(T)

Page 14: Heat and Mass Transfer Resistances

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

T

TX1CC o

AoA

'kk

C'kk'r

rc

ArcAs

gs

cm4242k

3

c

T

1

T

1

R

E

rorRe'kT'k

'rdW

dXF AAo

Mole BalanceMole Balance

Rate LawRate Law Energy BalanceEnergy Balance

io

piio

r TCF

HXT

Page 15: Heat and Mass Transfer Resistances

Multicomponent Diffusion

• Exact form of the flux equation for multicomponent mass transport:

• A simplified form uses a mean effective binary diffusivity,

1N,,2,1j , NyyDCNN

1kkj

1N

1kkjktj

N

1kkjjjmtj NyyDCN

Page 16: Heat and Mass Transfer Resistances

Multicomponent Diffusion

• The Stefan-Maxwell equations (Bird, Stewart, Lightfoot) are given for ideal gases:

• For binary system:

N

jk 1kkjjk

jkjt NyNy

D1

yC

211112

1t NNyND1

yC

Page 17: Heat and Mass Transfer Resistances

Multicomponent Diffusion

• Solved for flux

• Simplified forassumed equimolarcounter-diffusion

211112

1t NNyND1

yC

211112t1 NNyyDCN

112t1 yDCN

Page 18: Heat and Mass Transfer Resistances

Multicomponent Diffusion

• The effective binary diffusivity for species j can then be defined by equating the driving force terms of the expression containing Djm and the Stefan-Maxwell

N

jk 1kkjjk

jkjt NyNy

D1

yC

N

1kkjjmjtj NyDyCN

Page 19: Heat and Mass Transfer Resistances

Multicomponent Diffusion

• The effective binary diffusivity for species j can then be defined by equating the driving force terms of the expression containing Djm and the Stefan-Maxwell

N

1kkj

N

jk 1kkjjk

jkjmj NyNyNy

D1

DN

Page 20: Heat and Mass Transfer Resistances

Multicomponent Diffusion

• 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

1kkjj

N

jk 1kkjjk

jk

jm NyN

NyNyD1

D1

N

3,2k k1

k

1m1 Dy

y11

D1

Page 21: Heat and Mass Transfer Resistances

Multicomponent Diffusion• For reacting systems where steady-state flux ratios are

determined by reaction stoichiometry,

N

jk j

kjk

jkjjN

1k k

kj

N

jk j

kjk

jk

jm

yyD1

y11

y1

yyD1

D1

constantN

j

j

Page 22: Heat and Mass Transfer Resistances

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 PA

e

Page 23: Heat and Mass Transfer Resistances

Diffusion/Rxn in Porous Catalysts

• Extended Stefan-Maxwell

• Solved for binary, steady-state, 1D diffusion

Kj,e

Dj

N

1k

Dkj

Djk

jk,ej D

NNyNy

D1

pRT

1

KA,eAB,e0AAB

KA,eAB,eAAB

BA

AB,etA DDyNN11

DDLyNN11ln

NN1LDC

N

Page 24: Heat and Mass Transfer Resistances

Diffusion/Rxn in Porous Catalysts

• 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 D1

yyD

1D

1

Page 25: Heat and Mass Transfer Resistances

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

Page 26: Heat and Mass Transfer Resistances

Diffusion/Rxn in Porous Catalysts

steady state mass balance

rate in at r

r

2Ar r4W

rate out at r + r

rr

2Ar r4W

rate of generation within shell

c

mass catalyst

rate reaction

volume shell

mass catalyst volume shell

rr + r

R

rr4 2m'

Ar

0rr4r

r4Wr4W

2mC

'A

rr

2Arr

2Ar

0rr

dr

rWd 2C

'A

2Ar

BA cat

Page 27: Heat and Mass Transfer Resistances

Diffusion/Rxn in Porous Catalysts

0rr

dr

rWd 2C

'A

2Ar

dr

dCDW A

eAr

0rrrdr

dCD

dr

d 2C

'A

2Ae

0rSrrdr

dCD

dr

d 2Ca

"A

2Ae

2AnaCA

2Ana

'A

2An

"A

CkSr

CkSr

Ckr

0rSCk 2Ca

nAn

rate equationdefinitions

substitute Fick’s Law

Page 28: Heat and Mass Transfer Resistances

Diffusion/Rxn in Porous Catalysts

0rSCkrdr

dCD

dr

d 2Ca

nAn

2Ae

identify boundary conditions

finiteC0rA

symmetry

AsRrA CC

surface

dimensionless

As

A

C

C

R

r

AsA C

d

dC

R

1

d

dr

R

C

d

d

dr

dC AsA

0CD

Sk

dr

dC

r

2

dr

Cd nA

e

CanA2A

2

0D

CRSk

d

d2

d

d n

e

1nAs

2Can

2

2

Page 29: Heat and Mass Transfer Resistances

Diffusion/Rxn in Porous Catalysts

define Thiele modulus (n)

0D

CRSk

d

d2

d

d n

e

1nAs

2Can

2

2

e

1nAs

2Can2

n D

CRSk

0d

d2

d

d n2n2

2

understand the Thiele modulus

R0CD

RCSk

Ase

nAsCan2

n

reaction rate

diffusion rate

large n - diffusion controls

small n - kinetics control

Page 30: Heat and Mass Transfer Resistances

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

define y =

0d

d2

d

d 212

2

2

e

Can21 R

D

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

1A sinhcosh 11

differential has the solution apply boundary conditions

1 ,1

finite is ,0

Page 31: Heat and Mass Transfer Resistances

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

0d

d2

d

d 212

2

2

e

Can21 R

D

Sk

0yd

yd 212

2

1111 sinhBcoshAy

1B

1A sinhcosh 11

differential has the solution apply boundary conditions

1 ,1

finite is ,0

1

1

sinh

sinh1

As

A

C

C

Page 32: Heat and Mass Transfer Resistances

Thiele Modulus

As

A

C

C

Page 33: Heat and Mass Transfer Resistances

Internal Effectiveness Factor ()

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

sAs T ,C to exposed weresurface entire if rate

rate reaction overall actual

As

A"As

"A

'As

'A

As

A

M

M

r

r

r

r

r

r

M mol / timer mol / time / mass cat

Page 34: Heat and Mass Transfer Resistances

Internal Effectiveness Factor ()

• Determine MAs (rate if all surface at CAs) catalyst mass

catalyst mass

area surfacearea unit per rateMAs

'Asr

aS

CVAsM

x

x

As1Ck

c3

34

aAs1As RSCkM

Page 35: Heat and Mass Transfer Resistances

Internal Effectiveness Factor ()

• Determine MA (actual rate is equal to reactant diffusion rate at outer surface)

1AseA d

dCRD4M

11

12

1

11

1 sinh

sinh1

sinh

cosh

d

d

1coth 11

1cothCRD4M 11AseA

Page 36: Heat and Mass Transfer Resistances

Internal Effectiveness Factor ()

• Substitute results into definition of

As

A

M

M

c

334

aAs1

11Ase

RSCk

1cothCRD4

1cothRSk

D3 11

c2

a1

e

1coth3

1121

1coth3

1121

Page 37: Heat and Mass Transfer Resistances

Internal Effectiveness Factor ()

1coth3

1121

12

131

21

101

ac1

e21

SkD

R33

1 20

small dp

Page 38: Heat and Mass Transfer Resistances

Internal Effectiveness Factor ()

1coth3

1121

12

131

21

101

ac1

e21

SkD

R33

1 20

reactionrate

limited

internaldiffusionlimited

Page 39: Heat and Mass Transfer Resistances

Revisit and

• Thiele modulus - – Derived for spherical particle geometry– Derived for 1st order kinetics

• For large , approximately

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

21

213

1n

2

Page 40: Heat and Mass Transfer Resistances

Non-Isothermal Behavior

• For exothermic reactions, can be > 1 as internal temperature can exceed Ts.

• 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 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

Number ArrheniussRT

E

st

Aserxn

s

smax

Tk

CDH

T

TT

14

Page 41: Heat and Mass Transfer Resistances

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,C to exposed weresurface entire if rate

rate reaction overall actual

Page 42: Heat and Mass Transfer Resistances

Overall Effectiveness Factor

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

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

V1SarM cac"AA

b

bac"AcA SaraW

ba"AcA SraW

Page 43: Heat and Mass Transfer Resistances

Overall Effectiveness Factorba

"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 CAs and CA

defined by the as:

VSCkVaW baAs1cA

baAs1cAsbulk,Ac SCkaCCk

Page 44: Heat and Mass Transfer Resistances

Overall Effectiveness Factorba

"AcA SraW reaction rate

(internal & external surfaces)

VaCCkVaW cAsbulk,AccAr mass transport rate

As1"As

"A Ckrr Relation between CAs and CA

defined by the as:

ba1cc

bulk,AccAs Skka

CkaC

Solving for CAs:

Page 45: Heat and Mass Transfer Resistances

Overall Effectiveness Factorba

"AcA SraW reaction rate

(internal & external surfaces)

VaCCkVaW cAsbulk,AccAr mass transport rate

ba1cc

bulk,Acc1"A Skka

Cakkr

Substitution into the rate law:

ba1cc

bulk,AccAs Skka

CkaC

Solving for CAs:

Page 46: Heat and Mass Transfer Resistances

Overall Effectiveness Factorsummary of factor relationships:

ba1cc

bulk,Acc1"A Skka

Cakkr

Rearranging the expression:

bulk,A1ccba1

CkakSk1

"bulk,A

"A rr ccba1 akSk1

ccba1 akSk1

"As

"bulk,A

"A rrr

As1"As Ckr

Ab1"Ab Ckr

Overall Effectiveness Factor ()

Page 47: Heat and Mass Transfer Resistances

Weisz-Prater Criterion• Weisz-Prater Criterion is a method of determining if a given

process 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

Ase

c2'

obs,A21WP CD

RrC

Page 48: Heat and Mass Transfer Resistances

Mears’ Criterion

• Mass transfer effects negligible when it is true that

– n is the reaction order, and the transfer coefficients kc 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

15.0Ck

nRr

Abc

b'A

15.0

ThR

RE rH2bg

b'Arxn

Page 49: Heat and Mass Transfer Resistances

0rdz

dCU

dz

CdD A

'A

Ab2Ab

2

AB

Application to PBRs• Shell balance on

volume element Az

• Mole flux of A

• First order reaction

0rdz

dWb

'A

Az

UCdz

dCDW Ab

AABAz

Aba'Ab

'A CkSrr

0CkS Abab

Page 50: Heat and Mass Transfer Resistances

0CkSdz

dCU

dz

CdD Abab

Ab2Ab

2

AB

Application to PBRs

• Axial dispersion negligible (relative to forced axial convection) when…– dp is the particle diameter

– Uo is the superficial velocity of the gas

– Da is the effective axial dispersion coefficient

a

po

Abo

pb'A

D

dU

CU

dr

Which can be rewritten as:

Page 51: Heat and Mass Transfer Resistances

Application to PBRs

Which can be rewritten as:

AbabAb C

U

kS

dz

dC

Entrance condition:oAb0zAb CC

Integrating and applying boundary condtion yields:

U

zkSexpCC ab

AbAb o

U

zkSexpCC ab

AbAb o