kinetics of heterogeneous catalytic reactions reinaldo giudici

Kinetics of Heterogeneous Catalytic Reactions

UNIVERSIDADE DE SÃO PAULOESCOLA POLITÉCNICA

DEPARTAMENTO DE ENGENHARIA QUÍMICA

Reinaldo Giudici

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

KINETICS OF HETEROGENEOUS

CATALYTIC REACTIONS

Catalyst

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

B+C∆H

EcatalyzedEuncatalized

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

C(z,r)C(x)

ADSORPTION EQUILIBRIUM LANGMUIR ISOTHERM

Adsorption Isotherms

ASAdes

vAAads

Ckr Cpkr

+======

←→+

: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

ISOTHERM) ADSORPTION (LANGMUIR 1

Afor constant

mequilibriuadsorption

mequilibriu at the :desorption of rate :adsorption of rate

eqAAteqASeqASeqASteqAA

eqASeqvteqASeqveqAA

AeqveqA

eqASAeqveqAAdesads

ASAdes

vAAads

CCCCCpK

CkCpkrr Ckr Cpkr

+=⇒=−

=⇒===

←→+

: thatShow

:balance site00

B andA of adsorption ecompetitiv

eqBBeqAA

eqBSeqASeqvt

eqBSBeqveqBBB

eqASAeqveqAAA

adsdes

pKpKpK

CkCpkrCkCpkr

++==Θ

←→+

: thatShow

:balance site0

ondissociati with A of adsorption

eqASeqvt

eqASAeqveqAAA

CkCpkr

←→+

LHHW kinetic modelfor heterogeneous catalytic reactions

LHHW = Langmuir-Hinshelwood-Hougen-Watson

LHHW mechanism / kinetic model

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

steps)y (elementar Mechanism

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

−=+←

−=←

===−+←→

CkRSDDS

CkRSCCS

CCkRDSCSSAS

CpkRASSA

pppfrrrDCA

DSCSvAS

DCADCA

−=+←

−=←

===−+←→

DSCSvASsr

DSCSvAS

DCADCA

CpCkRSDDS

CpCkRSCCS

CCkKCC

CCkRDSCSSAS

CCpkRASSA

pppfrrrDCA

KKKkkk

DadsDdes

CadsCdes

======

−=+←

−=←

===−+←→

DSCSvASsr

DSCSvAS

DCADCA

CpCkRSDDS

CpCkRSCCS

CCkKCC

CCkRDSCSSAS

CCpkRASSA

pppfrrrDCA

RrRrRr

+=+=−=

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

=++−−==−+==−+==−+=

RRRRrRRrRRrRRr

−=+←

−=←

===−+←→

DSCSvASsr

DSCSvAS

DCADCA

CpCkRSDDS

CpCkRSCCS

CCkKCC

CCkRDSCSSAS

CCpkRASSA

pppfrrrDCA

RrRrRr

+=+=−=

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

=++−−==−+==−+==−+=

RRRRrRRrRRrRRr

4321 RRRR ===

DSCSASvt CCCCC +++=balance site

=→→→→→

−=+←

=→→→→→

−=+←

=→→

−=+←

=→→→→→

−=←

+←→

reaction global theof

constant mequilibriu

steplimiting

rate ,

...reaction global

KKppKK

CpKCpKCCpKkRR

CpKCCpKC

kRSDDS

CpKCCpKC

kRSCCS

CCCCkRDSCSSAS

CpKCKC

CpkRASSA

DCAvAsrglobal

DCDCAvAsr

vDDvCCvvAAsrglobal

vDDDSvDD

vCCCSvCC

globalsr

DSCSvASsr

vAAASA

pKp KpK

pCKkRR

pKp KpK CC CpKCp KCpK CC CCCC

CpKCCpKC

kRSDDS

CpKCCpKC

kRSCCS

CCCCkRDSCSSAS

CpKCKC

CpkRASSA

DDCCAA

DCAtAsr

global

DDCCAAvt

vDDvCCvAAvDSCSASvt

DCAvAsrglobal

vDDDSvDD

vCCCSvCC

globalsr

DSCSvASsr

vAAASA

steplimiting

rate ,

...reaction global

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

=→→→→→

−=+←

=→→→→→

−=+←

=→→

−=+←

=→→→→→

−=←

+←→

DDCCDCA

global

DDCCDCsr

DCvDSCSASvt

DCAvAvDC

DCvAAglobal

vDDDSvDD

vCCCSvCC

DSCSAS

DSCSvASsr

globalA

pKpKppK

KKCCCCCC

pCkCppKKKK

CpKCCpKC

kRSDDS

CpKCCpKC

kRSCCS

KKKCCC

CCCCkRDSCSSAS

CpkRASSA

+++=+++=

=→→→

−=+←

=→→→

−=+←

=→→→→→

−=←

+←→

limitingrate

...reaction global

+++=+++=

=→→→→

−=+←

=→→→→

−=+←

==→→

−=+←

=→→→

−=←

+←→

global

AsrAAvDSCSASvt

DvCvCv

AsrCglobal

vDDDSvDD

globalvCC

vASsrCS

DSCSvASsr

vAAASA

pKppKKpKp

pKCkRR

pKCCCCCC

ppKCkCpC

CpKCCpKC

kRSDDS

RRCpKC

kRSCCS

CCkRDSCSSAS

CpKCKC

CpkRASSA

limitingrate

...reaction global

+++=+++=

=→→→→

−=+←

=→→→→

−=+←

==→→

−=+←

=→→→

−=←

+←→

ADCCAAC

global

AsrCCAAvDSCSASvt

CvDvDv

AsrDglobal

globalvDD

vCCCSvCC

vASsrDS

DSCSvASsr

vAAASA

ppKKpKpKp

pKCkRR

pKpKCCCCCC

ppKCkCpC

RRCpKC

kRSDDS

CpKCCpKC

kRSCCS

CCkRDSCSSAS

CpKCKC

CpkRASSA

steplimiting

...reaction global

ADCCAAC

global

ppKKpKpKp

pKCkRR

global

pKppKKpKp

pKCkRR

DDCCDCA

global

pKpKppK

pKp KpK

pCKkRR

DDCCAA

DCAtAsr

global 2

LHHW mechanism

reaction global

DSCSSAS

+←→

+←→+

←→+

+←→

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

Other reactions

+←→+

←→+

+←→

←→

( )( )( )

group adsorptiongroup rcedriving_fofactor kinetic rate overall =

LHHW kinetic model

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

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

LHHW kinetic model

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

+←→+

←→+

←→

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).

steps)y (elementar Mechanism Rideal-Eley

...reaction global

−=+←

−=←

+←→

CkRSCCS

CkRDCSAS

CpkRASSA

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

one reaction & isothermal reactor• Batch reactor

• CSTR

• PFR

• PBR

V−−

dWdF ′=

A = )( Vrdt

dXN AA

Ao −=

AoAA r

−−

)(0 AA

dXF −=

)(0 AA

A rdWdXF ′−=

∫ −=

AAo Vr

∫ −=

∫ ′−=

Use of LHHW kinetic model in reactor design

)(0 AA

A rdWdXF ′−=

∫ ′−=

RTCRTCp

XXRTCRTCp

)1()1(

pKp KpK

DDCCAA

DCAtAsr

IoDoCoAo

. 0δε

ADCCAAC

global

ppKKpKpKp

pKCkRR

global

pKppKKpKp

pKCkRR

DDCCDCA

global

pKpKppK

pKp KpK

pCKkRR

DDCCAA

DCAtAsr

global 2

DCA +←→Discrimination of kinetic models using

initial rates (pC=pD=0)

AAglobal pkRR == 1

AAsrglobal 22 )1( +

Cglobal K

kRR == 3

Dglobal K

kRR == 4

Analysis of the rate equation

DDCCBBAA

CKCKCKCKCkCr

++++=−

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.)

ADCCAAC

global

ppKKpKpKp

pKCkRR

global

pKppKKpKp

pKCkRR

DDCCDCA

global

pKpKppK

pKp KpK

pCKkRR

DDCCAA

DCAtAsr

global 2

DCA +←→Discrimination of kinetic models using

initial rates (pC=pD=0)

AAglobalpp pkRRDC == → ==

AAsrglobal

pp DC22

)1( +== → ==

Cglobal

RRDC == → ==3

Dglobal

KkRRDC == → ==

DCA +←→

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

AAinitial pkR =

AAsrinitial 2)1( +

Cinitial K

Adsorption of A Surface reaction Desorption of C

DCA +←→

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

AAsrinitial 2)1( +

Surface reaction

AoAsrAo 2)1(

bXaYpkKkr

Ao +=⇔+=−

AopX =

∑ ∑

∑ ∑∑

−===

11slope

−=== ∑ ∑

xbynKk

11intercept abK A ==

interceptslope

1slope

pbaFYYbaF

i calciAo

icalcii ∑∑

−−

−=⇔−=

2,exp, )()(

),(min),(min

Linearization of the model equation

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

AoAsrAo 2)1(

)( AorY −=

AopX =

[ ] rrKkFi

calciAoiAoAsr ∑=

−−−=5

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?

• Model:

• Experimental data:

• General criterion:max likelihood function

Parameter Estimation

( ) ,y f x= β

( )∏=

iii xypxyL

,),(max βββ

),( tsmeasuremenn ii xy

Least Squares Criterion

( ) ( )p y x

y yi i

i i, exp

βπσ σ

=− −

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

( ) ( )p y x

y yi i

i i, exp

βπσ σ

=− −

( ) ( )

( ) ( ) ( ) ( )

−⇒

−−

⇒⇒

−−==

∑∑

∏∏

yyyyLL

yyxypxyL

constant

ˆminˆ2

1lnmax)ln(maxmax

σπσ

σπσββ

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)

( ) ( )

− ∑∑

iiii yyyyw

2 ˆ1minˆminσ

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

Error(y) =N(mean=0, σ2y)

ERROR-IN-VARIABLES

( ) ( )min 1 1

11 σ σyii i

y y x x− + −

==∑∑

( ) ( )p y x

y yi i

i i, exp

βπσ σ

=− −

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

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

= +β β0 1

( / ) ( ) ( )( )10 1

y a b xY X

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

= +β β0 1

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

y a b a xY X

= +β β

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

= +β β0 1

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

= + −β β0 1

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

xNon-symmetrical error distribution

Symmetrical error distribution

Variables transformation

Variable transformation changes the error distribution

16Y=20±4 20

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

0.0125

0.0083

kinetics of heterogeneous catalytic reactions reinaldo giudici

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