kinetics of heterogeneous catalytic reactions reinaldo giudici

Kinetics of Heterogeneous Catalytic Reactions

UNIVERSIDADE DE SÃO PAULOESCOLA POLITÉCNICA

DEPARTAMENTO DE ENGENHARIA QUÍMICA

Reinaldo Giudici

1

PQI-3401 – Engenharia de Reações Químicas II São Paulo, SP, janeiro/2021

KINETICS OF HETEROGENEOUS

CATALYTIC REACTIONS

2

Catalyst

3

Reaction Mechanism is a set of elementary reaction steps that explainshow the reactants are converted into the products.

Besides the reactants and products, the reaction mechanism includesformation and consumption of active intermediates

Typical characteristics of an active intermediate:- Highly reactive- Very short lifetime (e.g. ~10-9 s)- Difficult to measure, very low concentrations

Examples:- activated complex- transition state- free radicals- enzyme-substrate complex- ions

...

Catalysts• Change (increase) the reaction rate

– Reduce activation energy– Promote a new mechanism for the reaction

• Do not affect the reaction equilibrium• Not consumed• Can be selective for a given product • Can be homogeneous or heterogeneous

4

ener

gy

A

B+C∆H

EcatalyzedEuncatalized

A B + C, exothermicCatalyst (different intermediate) (different mechanism) Ecatalyzed < Euncatalyzed

A B

C(z,r)C(x)

ADSORPTION EQUILIBRIUM LANGMUIR ISOTHERM

6

Adsorption Isotherms

7

ASvt

t

AS

v

A

ASAdes

vAAads

des

ads

g

CCCC

C C

p

Ckr Cpkr

ASSA

+======

==

==

←→+

−

:BALANCE SITEcat) sites/g of (mol sites ofion concentrat surface total

cat) sites/g vacant of (mol vacant ofion concentrat surface cat) A/g (molA speciesby occupied sites ofion concentrat surface

(atm) phase gaseous in theA of pressure partial S site on the adsorbedA moleculeAS

sitevacant Sphase gaseousin A molecule A

:desorption of rate :adsorption of rate

)(


8

( )

ISOTHERM) ADSORPTION (LANGMUIR 1

1

and

Afor constant

mequilibriuadsorption

mequilibriu at the :desorption of rate :adsorption of rate

,

,,

,

,,,,,

,,,,,

,,

,

,,,

)(

eqAA

eqAA

t

eqASA

eqAA

eqAAteqASeqASeqASteqAA

eqASeqvteqASeqveqAA

AeqveqA

eqAS

A

A

eqASAeqveqAAdesads

ASAdes

vAAads

des

ads

g

pKpK

CC

pKpK

CCCCCpK

CCCCCpK

KCp

Ckk

CkCpkrr Ckr Cpkr

ASSA

+==Θ

+=⇒=−

+==

===

=⇒===

←→+

−

−

−


9

1

1

: thatShow

:balance site00

B andA of adsorption ecompetitiv

,,

,,

,,

,,

,,,

,,,

,,,

eqBBeqAA

eqBB

t

eqBSB

eqBBeqAA

eqAA

t

eqASA

eqBSeqASeqvt

eqBSBeqveqBBB

eqASAeqveqAAA

des

adsdes

ads

pKpKpK

CC

pKpKpK

CC

CCCC

CkCpkrCkCpkr

BSSB

ASSA

++==Θ

++==Θ

++=

=−=

=−=

←→+

←→+

−

−

1

: thatShow

:balance site0

22

ondissociati with A of adsorption

,2

,2,

,,

2,

2,,22

2

2

eqAA

eqAA

t

eqASA

eqASeqvt

eqASAeqveqAAA

des

ads

pK

pKC

C

CCC

CkCpkr

ASSA

+==Θ

+=

=−=

←→+

−

LHHW kinetic modelfor heterogeneous catalytic reactions

LHHW = Langmuir-Hinshelwood-Hougen-Watson

12

LHHW mechanism / kinetic model

13

Hypotheses:

(1) in the surface reaction step, all reactants areadsorbed and all products formed are adsorbedon the catalytic sites

(2) PSSH applies for the reaction intermediates

(3) A rate-limiting step is assumed (i.e., one of thereaction steps is rate-limiting)

LHHW mechanism/kinetic model

14

)4(

)3(

)2(

)1(

steps)y (elementar Mechanism

),,()( ......... ...reaction global

444

4

333

3

222

2

111

1

−=+←

→

−=+←

→

−=+←

→+

−=←

→+

===−+←→

KCp

CkRSDDS

KCp

CkRSCCS

KCC

CCkRDSCSSAS

KC

CpkRASSA

pppfrrrDCA

vDDS

vCCS

DSCSvAS

ASvA

DCADCA


−=

−=+←

→

−=

−=+←

→

−=

−=+←

→+

−=

−=←

→+

===−+←→

vDD

DSD

vDDS

vCC

CSC

vCCS

sr

DSCSvASsr

DSCSvAS

A

ASvAA

ASvA

DCADCA

CpKC

kK

CpCkRSDDS

CpKC

kK

CpCkRSCCS

KCC

CCkKCC

CCkRDSCSSAS

KC

CpkK

CCpkRASSA

pppfrrrDCA

)4(

)3(

)2(

)1(



444

4

333

3

222

2

111

1

sr

sr

AAads

AAads

KKkk

KKKkkk

==

==

==

2

2

,1

,1

D

D

D

DadsDdes

DDads

Ddes

C

C

C

CadsCdes

CCads

Cdes

Kk

Kk

kkKk

kkkK

Kk

Kk

kkKk

kkk

K

======

======

−

−

,,4

,

,

4

44

,,3

,

,

3

33

1

1


−=

−=+←

→

−=

−=+←

→

−=

−=+←

→+

−=

−=←

→+

===−+←→

vDD

DSD

vDDS

vCC

CSC

vCCS

sr

DSCSvASsr

DSCSvAS

A

ASvAA

ASvA

DCADCA

CpKC

kK

CpCkRSDDS

CpKC

kK

CpCkRSCCS

KCC

CCkKCC

CCkRDSCSSAS

KC

CpkK

CCpkRASSA

pppfrrrDCA

)4(

)3(

)2(

)1(



444

4

333

3

222

2

111

1

4

3

1

RrRrRr

D

C

A

+=+=−=

(PSSH) 0(PSSH) 0(PSSH) 0(PSSH) 0

4321

42

32

21

=++−−==−+==−+==−+=

RRRRrRRrRRrRRr

S

DS

CS

AS

????

====

v

DS

CS

AS

CCCC


−=

−=+←

→

−=

−=+←

→

−=

−=+←

→+

−=

−=←

→+

===−+←→

vDD

DSD

vDDS

vCC

CSC

vCCS

sr

DSCSvASsr

DSCSvAS

A

ASvAA

ASvA

DCADCA

CpKC

kK

CpCkRSDDS

CpKC

kK

CpCkRSCCS

KCC

CCkKCC

CCkRDSCSSAS

KC

CpkK

CCpkRASSA

pppfrrrDCA

)4(

)3(

)2(

)1(



444

43

33

32

22

21

11

1

4

3

1

RrRrRr

D

C

A

+=+=−=

(PSSH) 0(PSSH) 0(PSSH) 0(PSSH) 0

4321

42

32

21

=++−−==−+==−+==−+=

RRRRrRRrRRrRRr

S

DS

CS

AS

4321 RRRR ===

DSCSASvt CCCCC +++=balance site

==

=

=

−=

=

−=

−==

=→→→→→

−=+←

→

=→→→→→

−=+←

→

=→→

−=+←

→+

=→→→→→

−=←

→+

+←→

reaction global theof

constant mequilibriu

1

steplimiting

rate ,

...reaction global

2

22

4

4

3

3

22

2

1

1

KKpp

p

CCCC

CpC

CpC

CpC

KKKK

Kpp

pCKkR

KKppKK

pCKkK

CpKCpKCCpKkRR

CpKCCpKC

kRSDDS

CpKCCpKC

kRSCCS

RRK

CCCCkRDSCSSAS

CpKCKC

CpkRASSA

DCA

eqDC

A

eqvAS

DSCS

eqvA

AS

eqvD

DS

eqvC

CS

srA

DC

DCAvAsrglobal

srA

DCDCAvAsr

sr

vDDvCCvvAAsrglobal

vDDDSvDD

DSD

vCCCSvCC

CSC

globalsr

DSCSvASsr

vAAASA

ASvAA

pKp KpK

Kpp

pCKkRR

pKp KpK CC CpKCp KCpK CC CCCC

Kpp

pCKkR

CpKCCpKC

kRSDDS

CpKCCpKC

kRSCCS

RRK

CCCCkRDSCSSAS

CpKCKC

CpkRASSA

DCA

DDCCAA

DCAtAsr

global

DDCCAAvt

vDDvCCvAAvDSCSASvt

DCAvAsrglobal

vDDDSvDD

DSD

vCCCSvCC

CSC

globalsr

DSCSvASsr

vAAASA

ASvAA

2

2

2

2

4

4

3

3

22

2

1

1

)1(

)1(

steplimiting

rate ,

...reaction global

+++

−

==

+++=+++=+++=

−=

=→→→→→

−=+←

→

=→→→→→

−=+←

→

=→→

−=+←

→+

=→→→→→

−=←

→+

+←→

DDCCDCA

DCAtA

global

DDCCDCsr

DCvDSCSASvt

DCAvAvDC

srA

DCvAAglobal

vDDDSvDD

DSD

vCCCSvCC

CSC

vDCsr

DC

srv

DSCSAS

sr

DSCSvASsr

globalA

ASvAA

pKpKppK

KKpp

pCkRR

pKpKppK

KKCCCCCC

Kpp

pCkCppKKKK

CpkRR

CpKCCpKC

kRSDDS

CpKCCpKC

kRSCCS

CppK

KKKCCC

CK

CCCCkRDSCSSAS

RRKC

CpkRASSA

DCA

+++

−

==

+++=+++=

−=

−==

=→→→

−=+←

→

=→→→

−=+←

→

==→

−=+←

→+

=→→→→→

−=←

→+

+←→

1

1

,

step

limitingrate

...reaction global

1

1

4

4

3

3

2

2

11

1

+++

−

==

+++=+++=

−=

−==

=→→→→

−=+←

→

=→→→→

−=+←

→

==→→

−=+←

→+

=→→→

−=←

→+

+←→

DDD

ACAAD

DCAtC

global

DDD

A

D

AsrAAvDSCSASvt

DCA

DvCvCv

D

A

DC

AsrCglobal

vDDDSvDD

DSD

globalvCC

CSC

vDD

AAsr

DS

vASsrCS

sr

DSCSvASsr

vAAASA

ASvAA

pKppKKpKp

Kpp

pKCkRR

pKpp

KKK

pKCCCCCC

Kpp

ppKCkCpC

pp

KKKK

kRR

CpKCCpKC

kRSDDS

RRCpKC

kRSCCS

CpK

pKKC

CCKC

KCC

CCkRDSCSSAS

CpKCKC

CpkRASSA

DCA

1

1

step

limitingrate

,

...reaction global

3

3

4

4

33

3

2

2

1

1

++++

−

==

+++=+++=

−=

−==

=→→→→

−=+←

→

=→→→→

−=+←

→

==→→

−=+←

→+

=→→→

−=←

→+

+←→

C

ADCCAAC

DCAtD

global

C

A

C

AsrCCAAvDSCSASvt

DCA

CvDvDv

C

A

DC

AsrDglobal

globalvDD

DSD

vCCCSvCC

CSC

vCC

AAsr

CS

vASsrDS

sr

DSCSvASsr

vAAASA

ASvAA

ppKKpKpKp

Kpp

pKCkRR

pp

KKK

pKpKCCCCCC

Kpp

ppKCkCpC

pp

KKKK

kRR

RRCpKC

kRSDDS

CpKCCpKC

kRSCCS

CpK

pKKC

CCKC

KCC

CCkRDSCSSAS

CpKCKC

CpkRASSA

DCA

1

1

steplimiting

rate

,

...reaction global

4

4

44

4

3

3

2

2

1

1

++++

−

==

C

ADCCAAC

DCAtD

global

ppKKpKpKp

Kpp

pKCkRR

14

+++

−

==

DDD

ACAAD

DCAtC

global

pKppKKpKp

Kpp

pKCkRR

13

DDCCDCA

DCAtA

global

pKpKppK

KKpp

pCkRR

+++

−

==1

1

pKp KpK

Kpp

pCKkRR

DDCCAA

DCAtAsr

global 2

2

2 )1(

+++

−

==

LHHW mechanism

reaction global

4

3

2

1

SDDS

SCCS

DSCSSAS

ASSA

DCA

+←→

+←→

+←→+

←→+

+←→

Results for LHHW(depending on therate-limiting step Chosen)


24

Other reactions

etc

.

.

.

.

DCBA

CBA

DCA

CA

+←→+

←→+

+←→

←→


25

( )( )( )

group adsorptiongroup rcedriving_fofactor kinetic rate overall =

LHHW kinetic model

26Froment & Bischoff, Chemical Reactor Analysis and Design, 2nd ed., Wiley, 1990

27

Froment & Bischoff, Chemical Reactor Analysis and Design, 2nd ed., Wiley, 1990

LHHW kinetic model

28


LHHW kinetic model

29



30

ASSIGNMENT: - choose one of these reactions;- write the steps of the LHHW mechanism;- choose one of the steps as rate-limiting;- derive the corresponding rate equation;- obtain the rate equation from the

tables (kinetic factor, driving-force group, adsorption group) and compare with the per-you-derived rate equation

.

.

.

DCBA

CBA

CA

+←→+

←→+

←→

31

Eley-Rideal mechanismSimilar to LHHW, except that in the surface reaction step, not all reactants (or not all the products) are adsorbed (i.e., hypothesis (1) of LHHW is not adopted).

)3(

)2(

)1(

steps)y (elementar Mechanism Rideal-Eley

...reaction global

333

3

222

2

111

1

−=+←

→

−=+←

→

−=←

→+

+←→

KCp

CkRSCCS

KpC

CkRDCSAS

KC

CpkRASSA

DCA

vCCS

DCSAS

ASvA

Design Equations for Ideal Reactors(mole balance for component A – limiting reagent)

one reaction & isothermal reactor• Batch reactor

• CSTR

• PFR

• PBR

)(0

A

AA

rFF

V−−

=

AA r

dWdF ′=

AA r

dVdF

=

Vrdt

dNA

A = )( Vrdt

dXN AA

Ao −=

)(0

A

AoAA r

XXFV

−−

=

)(0 AA

A rdV

dXF −=

)(0 AA

A rdWdXF ′−=

)(

0

∫ −=

A

A

X

X A

AAo Vr

dXNt

)(

0

∫ −=

A

A

X

X A

AAo r

dXFV

)(

0

∫ ′−=

A

A

X

X A

AAo r

dXFW

32

Use of LHHW kinetic model in reactor design

33

)(0 AA

A rdWdXF ′−=

)(0

∫ ′−=

A

A

X

X A

AAo r

dXFW

+

+

=

+

+

==

+

+

=

+

+

==

+−

=

+−

==

o

o

AA

AAo

Do

Aoo

o

AA

AAo

Do

AoDD

o

o

AA

AAo

Co

Aoo

o

AA

AAo

Co

AoCC

o

o

AA

AAo

o

o

AA

AAoAA

TPPT

X

XFF

PyTP

PTX

XFF

RTCRTCp

TPPT

X

XFF

PyTP

PTX

XFF

RTCRTCp

TPPT

XXPy

TPPT

XXRTCRTCp

)1()1(

)1()1(

)1()1(

)1()1(

εε

εε

εε

pKp KpK

Kpp

pCKkr

DDCCAA

DCAtAsr

A 2

2

)1(

)'(

+++

−

=−

+++

+=

==

IoDoCoAo

Ao

AA

FFFFF

y

)1(

. 0δε

++++

−

==

C

ADCCAAC

DCAtD

global

ppKKpKpKp

Kpp

pKCkRR

14

+++

−

==

DDD

ACAAD

DCAtC

global

pKppKKpKp

Kpp

pKCkRR

13

DDCCDCA

DCAvA

global

pKpKppK

KKpp

pCkRR

+++

−

==1

1

pKp KpK

Kpp

pCKkRR

DDCCAA

DCAtAsr

global 2

2

2 )1(

+++

−

==

DCA +←→Discrimination of kinetic models using

initial rates (pC=pD=0)

AAglobal pkRR == 1

pKpKk

RRAA

AAsrglobal 22 )1( +

==

C

Cglobal K

kRR == 3

D

Dglobal K

kRR == 4

Analysis of the rate equation

( )

4321

1

1)(

2/1

2

XD

D

XC

C

XB

B

XA

A

Y

A

BA

DDCCBBAA

BAA

Ck

KCk

KC

kKC

kK

krCC

CKCKCKCKCkCr

++++=

−

++++=−

35

Linear regression (using the linearized-transformed equation)

Nonlinear regression (using the original rate equation)


Recommended Exercises (Fogler Chapter 10)

P-10-5 (3ª. Ed.) = P-10-6 (4ª. Ed.)P-10-8 (3ª. Ed.) = P-10-9 (4ª. Ed.)

P-10-9 (3ª. Ed.) = P-10-10 (4ª. Ed.) P-10-10 (3ª. Ed.) = P-10-11 (4ª. Ed.)

36

++++

−

==

C

ADCCAAC

DCAtD

global

ppKKpKpKp

Kpp

pKCkRR

14

+++

−

==

DDD

ACAAD

DCAtC

global

pKppKKpKp

Kpp

pKCkRR

13

DDCCDCA

DCAtA

global

pKpKppK

KKpp

pCkRR

+++

−

==1

1

pKp KpK

Kpp

pCKkRR

DDCCAA

DCAtAsr

global 2

2

2 )1(

+++

−

==

DCA +←→Discrimination of kinetic models using

initial rates (pC=pD=0)

AAglobalpp pkRRDC == → ==

10

pKpKk

RRAA

AAsrglobal

pp DC22

0

)1( +== → ==

C

Cglobal

pp

Kk

RRDC == → ==3

0

D

Dglobal

pp

KkRRDC == → ==

40

DCA +←→

Discrimination of kinetic models using initial rates (pC=pD=0)

AAinitial pkR =

pKpKk

RAA

AAsrinitial 2)1( +

=C

Cinitial K

kR =

Adsorption of A Surface reaction Desorption of C

DCA +←→

Discrimination of kinetic models using initial rates (pC=pD=0)

pKpKk

RAA

AAsrinitial 2)1( +

=

Surface reaction

41

pKpKk

rAoA

AoAsrAo 2)1(

)(+

=−

bXaYpkKkr

p

XAo

bsr

a

Asr

Y

Ao

Ao +=⇔+=−

11)(

)( Ao

Ao

rp

Y−

=

AopX =

∑ ∑

∑ ∑∑

= =

= ==

−

−===

n

i

n

iii

n

i

n

ii

n

iiii

sr xn

x

xyn

xy

kb

1

2

1

2

1 11

1

11slope

−=== ∑ ∑

= =

n

i

n

iii

Asr

xbynKk

a1 1

11intercept abK A ==

interceptslope

bksr

1slope

1==

[ ] r

pr

pbaFYYbaF

i calciAo

Ao

iAo

Ao

icalcii ∑∑

==

−−

−=⇔−=

5

1

2

,exp,

5

1

2,exp, )()(

),(min),(min

Linearization of the model equation

Linear Regression (fit of the linearized equation to the transformed variables Y and X)

42

pKpKk

rAoA

AoAsrAo 2)1(

)(+

=−

)( AorY −=

AopX =

[ ] rrKkFi

calciAoiAoAsr ∑=

−−−=5

1

2,exp, )()(),(min

Fit of the original (nonlinear) equation to the original (non-transformed) data

Nonlinear Regression (fit of the nonlinear equation to the original data)

MatlabOctaveScilabPolymathExcel (“solve”)etc.

Try the two methods and compare• Linear regression using the linerized equation and

transformed variables

• Nonlinear regression using the original equation and the original variables

Which of the two methods above is the easiest one?Which of the two methods above is the best one?

43

• Model:

• Experimental data:

• General criterion:max likelihood function

Parameter Estimation

( ) ,y f x= β

( )∏=

=n

iii xypxyL

1

,),(max βββ

),( tsmeasuremenn ii xy

44

Least Squares Criterion

( ) ( )p y x

y yi i

i i, exp

βπσ σ

=− −

12 22

2

2

Assumptions:(1) Errors on variable x are zero

(i.e., x are “perfect” measurements, with no error)(2) Errors on the variable y is

distributed according to a normal (Gaussian) distribution with mean=zero and

(3) variance σ2 = constant

y

x45


( ) ( )p y x

y yi i

i i, exp

βπσ σ

=− −

12 22

2

2

( ) ( )

( ) ( ) ( ) ( )

−⇒

−−

⇒⇒

−−==

∑∑

∏∏

==

==

n

iii

n

iii

n

n

i

iin

iii

yyyyLL

yyxypxyL

1

2

1

22

constant

2

12

2

21

ˆminˆ2

12

1lnmax)ln(maxmax

2ˆ

exp2

1,),(

σπσ

σπσββ





46

y

x

Other Criteria(a) Error(x) = 0 and

Error(y) = N(mean=0, σ2 not constant)WEIGHTED LEAST SQUARES(with weights proportional to the inverse of the variance)

( ) ( )

−=

− ∑∑

==

n

iii

i

n

iiii yyyyw

1

22

1

2 ˆ1minˆminσ

(b) Error(x) = N(mean=0, σ2x) (not zero)

Error(y) =N(mean=0, σ2y)

ERROR-IN-VARIABLES

( ) ( )min 1 1

22

22

11 σ σyii i

xii i

i

n

i

n

y y x x− + −

==∑∑

y

x47


( ) ( )p y x

y yi i

i i, exp

βπσ σ

=− −

12 22

2

2





y

x48

Linearization of the equation by variable transformation

y ax xb c= 1 2 ln( ) ln( ) ( )ln( ) ( )ln( )y a b x c xY X X

= + +β β β0 1 1 2 2

1 2

y aebx= ln( ) ln( ) ( )( )y a b xY X

= +β β0 1

ya bx

=+1

( / ) ( ) ( )( )10 1

y a b xY X

= +

β β

y abx

= + ( ) ( ) ( )( / )y a b xY X

= +β β0 1

1

yax

b x=

+

( / ) ( / ) ( / )( / )1 1 10 1

y a b a xY X

= +β β

( / ) ( / ) ( / )( )x y b a a xY X

= +β β0 1

1

( ) ( ) ( )( / )y a b y xY X

= + −β β0 1

49

Linearization of the equation by variable transformationDisadvantages(1) If the values of x1 are in a narrow range (less than one order of

magnitude), ln(x1) will be in an even narrower range. Because allelements of 1st column of matrix X are equal to 1, the matrix XTX becomes quase-singular.

(2) Variable transformation may change the error structure (i.e., erroron the transformed variable is not normally distributed), thusintroducing bias in the parameter estimates.

(3) In some transformations, y enters in the transformed“independent” variables (thus adding errors to the tranformed X)

Advantages(a) Convert the original problem (nonlinear regression) into an easier

problem (linear regression)(b) Useful for obtaining the initial guesses of the parameters to be

used in the nonlinear regression. 50

y

x

y

xNon-symmetrical error distribution

Symmetrical error distribution

Variables transformation

51

Variable transformation changes the error distribution

16Y=20±4 20

24

ln(16) =2.773ln(Y)=2.996 ± ? ln(20) =2.996

ln(24) =3.178

1/16 = 0.06251/Y = 0.05 ± ? 1/20 = 0.0500

1/24 = 0.0417

0.223

0.182

0.0125

0.0083

4

4

kinetics of heterogeneous catalytic reactions reinaldo giudici

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