Reactor design equation - Packed bed -...

Catalysis Engineering - Kinetics

Catalytic Reaction Kinetics

Why Catalytic Reaction Kinetics ?Derivation rate expressionsSimplifications– Rate determining step– Initial reaction rate

Limiting cases– Temperature dependency– Pressure dependency

Examples


Reactor design equation - Packed bed

ν η= ⋅ ⋅ii W

dF rdW

stoichiometric coefficient i

catalyst effectiveness

rate expression

Molar flow i

‘bed coordinate’

Dimension of rate?


Rate expression

A + B C + DCatalyst

How much detail ?

A + B + * AB* * + C + D

A + B [intermediates] C + D

etc.

overall rate expression?

.....),,,,( catalystKkTpfr eqi=

conditions

rate constant(s) thermodynamics


Rate expression – Gas phase reaction

2CBA ckcckr ⋅−⋅⋅= −+

forward ratebackwardchance to success

amount of A × chance to meet B


Role of catalyst?

Concentrating reactantsadsorption/complexation

Providing alternative reaction pathcatalyst selectivityother activation energy barrier

But:– other components adsorb, too

block ‘active sites’– fixed number of ‘active sites’

rate constant

affect rate

affect rate

affect rate

other form rate expression expectedother form rate expression expectedCatalysis Engineering - Kinetics

Rate vs. pressure/concentration ?

ratemol/s.gcat

pressure / kPa

• fixed number of active sites• mono & bimolecular reactions


Rate expression – Catalysed reaction

CadsCB

adsA scksckr θθ ⋅⋅−⋅⋅= −+

forward ratebackwardchance to success

amount of A adsorbed

chance of adjacent B adsorbed

Note:• cgas and cads differ• ratios components differ


Kinetics

Rate expressions in principle crucial for– design– process start-up and control– process development and improvement– selection reaction model

Often used in catalysis– power rate models– models based on elementary processes

• extrapolation more reliable• intellectually process better understood

mj

ni ppkr ⋅⋅=

[ ]( )2

222

1/

BBAA

eqBABAT

pKpKKppKKksN

r++

−=


Simple example: reversible reactionA B

A B

A* B*

‘Elementary processes’

‘Langmuir adsorption’

monomoleculare.g. isomerization


Elementary processes

Rate expression follows from rate equation

r r r k p N k NA T T A= − = −− −1 1 1 1θ θ*

r r r k N k NT A T B= − = −− −2 2 2 2θ θ

r r r k N k p NT B B T= − = −− −3 3 3 3θ θ *

Eliminate unknown surface occupancies


Site balance:(3.5)

Steady state assumption:(3.6-7)

Rate expression:(3.9)

Elementary processes contd.

1= + +θ θ θ* A B

0

0

=

=

dtddt

d

B

A

θ

θ

321

321

:with(......)(......)(.....)

)/(

KKKK

ppKppkkkN

r

eq

BA

eqBAT

=

++−

=

MicrokineticsMichaelis-Menten

Algebraic eqs.


Quasi-equilibrium / rate determining step

r1

r2

r3

r-1

r-2

r-3

r

rate determining

‘quasi-equilibrium’

r = r2 - r-2


Rate expression - r.d.s.

r r r k N k NT A T B= − = −− −2 2 2 2θ θ

Rate determining step:

Eliminate unknown occupancies

Quasi-equilibrium:r r k p N k NA T T A1 1 1 1= =− − θ θ*

So: θ θ

θ θ

A A

BB

K p K kk

pK

= ⋅ =

= ⋅

−1 1

1

1

3

*

*

with:


Rate expression, contd.

Substitution:

{ }r r r k N K p k N p Kr k N K p k p k K K

T A T B

T A B

= − = −

= −− −

−

2 2 2 1 2 3

2 1 2 2 1 3

θ θθ

* *

*

//

K K K K ppeq

B

A eq

= =

1 2 3where:

Unknown still θ*


Rate expression, contd.

Site balance:

{ }1 1 1 3= + + = ⋅ + +θ θ θ θ* * /A B A BK p p K

( )θ * /=

+ +1

1 1 3K p p KA B

Finally:

{ }( )

rk N K p p K

K p p KT A B eq

A B

=−

+ +2 1

1 31/

/


Surface occupancies

Empty sites

Occupied by A

Occupied by B

( )θ BB

A B

p KK p p K

=+ +

//

3

1 31

( )θ AA

A B

K pK p p K

=+ +

1

1 31 /

( )θ * /=

+ +1

1 1 3K p p KA B

BKK

=3

1


Other steps rate determining

Adsorption r.d.s

Surface reaction r.d.s.

Desorption r.d.s.

{ }( )r

k N K p p KK p p K

T A B eq

A B

=−

+ +2 1

1 31/

/

{ }( )r

k N K K p p KK K p

T A B eq

A

=−

+ +3 1 2

2 11 1/

{ }( )r

k N p p KK p K

T A B eq

B

=−

+ +1

2 31 1 1/

/ /

Rule of thumb:Generally surface reaction r.d.s.Catalysis Engineering - Kinetics

Thermodynamics

Equilibrium constant

Adsorption constant

Reaction entropy

Reaction enthalpy

Adsorption enthalpy,<0(J/mol)

Adsorption entropy, <0(J/mol K)

atm-1

oooeq STTHTGKRT ∆−∆=∆= )()(ln

)(, TGi

oifi∑ ∆= ν

RTH

RSK

oo

A∆

−∆

=ln

Data sources: Handbooks, API,JANAFChemsage, HSC


Langmuir adsorption

Uniform surface (no heterogeneity)Discrete number of sitesNo interaction between adsorbed species

A + * A*

AA

AAA pK

pK+

=1

θ

KA /bar -1

0.1

1.0

10100

pA /bar

1.0

0.8

0.6

0.4

0

0.2

0 0.2 0.4 0.6 0.8 1.0


Multicomponent adsorption / inhibition

Langmuir adsorption

( )θ AA

A i i

K pK p K p

=+ + ∑

1

11

Inhibitors


Langmuir adsorption model

Generally used– although nonlinear, mathematically simple– simple physical interpretation– rather broadly applicable

• multicomponent adsorption• non-uniform surfaces

– ‘compensation effect’– very weak and strong sites do not contribute much

to the rate• for microporous media (activated carbons) often

not satisfactory


Dissociative adsorption

H2 + 2* 2H*

( )( )

θHH H

H H

K p

K p=

+2 2

2 2

0.5

0.51

Two adjacent sites needed


Initial rate expressions

Forward ratesProduct terms negligible

r k p A= '0 r k p

K pA

A A=

+

'0

01 r k= '

Adsorption Surface Desorption

r0

T1

T2

T3

pApA pA

T1

T2

T3

T1

T2

T3

See tutorial Catalysis Engineering - Kinetics

Dual site reaction :A + B C

A + * ↔ A*

B + * ↔ B*

A* + B* ↔ C* + *

C* ↔ C + *

(r.d.s.)

{ }{ }2

421

213

/1/KppKpKKpppKKNsk

rCBA

eqCBAT

+++

−=


Dual site reaction, contd.

{ }*3333 θθθθ CBAT kkNsrrr −− −⋅=−=

Number of neighbouring sites (here: 6)


More than one reactant

• One-site models

• Two-site models

e.g. hydrogenation, oxidation

dual site reaction

single site reaction

• Number of sites conditions dependent

)1(1 BABAA

BAABATABT pKpK

ppKKkNkNr++

== θ

2)1( BBAA

BBAATBAT pKpK

pKpKskNskNr++

== θθ

)1()1( BBAA

BBAATBAT pKpK

pKpKskNskNr++

== •∗θθ

( )( ){ }21

210

1 AA

AATT pK

pKNN+

=

different sites

optimal surface concentrationsoptimal adsorption strengths


Langmuir-Hinshelwood/Hougen-Watson models (LHHW)

r kinetic factor driving fo rceadsorp tion te rm

=⋅( ) ( )

( )n

includes NT, k(rds)For: A+B C+D

pApB-pCpD/Keq

molecular: KApAdissociative: (KApA)0.5 = 0, 1, 2...

number species inand before r.d.s.


What about observed: reaction orderactivation energy ?

Determination:

ii p

rnlnln

∂∂

=

ln r

ln pi

slope = order ni

ln r

1/T

slope = -Eaobs/R

TrRTE obs

a ∂∂

=ln2

LHHW models ? ( )r k N K p

K p K pT A A

A A B B

=+ +

2

1


Reaction order - Activation energy

( )r k N K p

K p K pT A A

A A B B

=+ +

2

1rate expression

Reaction order

Activation energy

BB

AA

nn

θθ

−=

−= 1

( ) BBAAaobsa HHEE ∆−∆−+= θθ12

• depend strongly on occupancy!• vary during reaction

limiting cases?


Limiting cases - forward ratesSurface reaction r.d.s. ( )

r k N K pK p K p

T A A

A A B B

=+ +

2

1

1. Strong adsorption Ar k NT= 2

E Eaobs

a= 2

A*

B*

A*#

Ea2


Limiting cases - forward rates

2. Weak adsorptionr k N K pT A A= 2

Surface reaction r.d.s. ( )r k N K p

K p K pT A A

A A B B

=+ +

2

1

E E Haobs

a A= +2 ∆

A*

A(g) + *

A*#

Ea2


Limiting cases - forward rates

3. Strong adsorption B

r k N K pK p

T A A

B B

= 2

Surface reaction r.d.s. ( )r k N K p

K p K pT A A

A A B B

=+ +

2

1

E E H Haobs

a A B= + −2 ∆ ∆

A*#

B* + A

B+*+AEa2

A*


Cracking of n-alkanes over ZSM-5J. Wei I&EC Res.33(1994)2467

AA pKkr 20 =

Aaobsa HEE ∆+= 2

Carbon number

negative!?Ea2

Eaobs

∆HA

kJ/mol

200

100

-200

-100

0

0 5 10 15 20


n-Alkanes cracking

Energy diagram

Initial state Transition state

Adsorbed state

A + *

∆HadsEa2

Ea,obs

A*B*,C*


Observed temperature behaviour

• T higher coverage lower• Highest Ea most favoured

Change in r.d.s.

1/T

ln robs

desorption r.d.s.

adsorption r.d.s.


Catalytic reaction kinetics, summary

Langmuir adsorption– uniform sites, no interaction adsorbed species,

finite number of sites, multicomponentRate expression derivation– series of elementary steps– steady state assumption, site balance – quasi-equilibrium / rate determining step(s)– initial rates (model selection)

LHHW models– inhibition, variable reaction order, activation

energy

simpler

mechanism kinetics


Hydrodesulphurization kineticsSie, AIChE-J 42(1996)3498

• Apparent second order behaviour• H2S inhibits strongly

Example HDS vacuum gasoil

0.0 0.1 0.2 0.3 0.4 0.5 0.6

1/LHSV (h)

0.00.20.40.60.81.01.21.41.61.82.02.2

1/S-

1/S

0(1

/wt.%

)

GasoilCoMo-aluminatrickle flowL=0.2-0.4 m

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

bed length

c

concentration

conversion

fast decrease followedby slow decrease


Composition oil fractions

S

S

SR R

S

RS S

R

RS

R

0 5 10 15 20 25 30

Vacuum gasoil

Simulated distillation b.p.

Sulphur compounds

Thioethers

ThiopheneBenzthiophene

Dibenzthiophene

Substituteddibenzthiophene

complex mixturesdifferent reactivities⇒lumping


Simulated profiles - HDS reactivity lumping

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

bed length

conc

entra

tion

Three lump model: first order reactionsSimulated model data:2nd orderk=10 m3/mol.sc0=2 mol/m3

Three lump model:1st ordersk1=36.1 s-1 c01=1.23k2=16.0 s-1 c02=0.59k3=7.5 s-1 c03=0.18

sum

1

23

Three lump model adequateInhibition through LHHW models Which groups lumped?

⇒model studies


Kinetic coupling between catalytic cycles

Bifunctional catalysis: Reforming

Isomerization n-pentane: n-C5 -> i-C5

Pt-function: n-C5 -> n-C5=

surface diffusionAcid function: n-C5= -> i-C5=

surface diffusionPt-function: i-C5= -> i-C5

Coupled catalytic cycles on different sites

low concentrationclose proximity

See tutorial


Initial rates - CO hydrogenation over Rh

Van Santen et al.

Kinetic model

1. CO + * ↔ CO*2. CO* + * → C* + O* (r.d.s.)

r sN kT CO0 2= ⋅θ θ *

{ }r sk N K p

K pT CO CO

CO CO0

221

=+

Initial rate

0.20.4

0.60.8

1.0

Occupancy (-)400

450500

550600

Temperature (K)

0

200

400

600

800

Rat

e



Catalysed N2O decomposition over oxidesWinter, Cimino

Rate expressions:

( )( )rk p

p Kobs N O

O

=+

2

21 3

0.5

r k pobs N O= ⋅ 2

( )r k p

pobs

N O

O

= ⋅ 2

2

0.5

1st order

strong O2 inhibition

moderate inhibition

Also: orders 0.5-1water inhibition

= Explain / derive =


N2O decomposition over Mn2O3

2 N2O 2N2 + O2

Vannice et al. J.Catal. 1996

Rate expression

( )( )r k N K pK p p K

T N O

N O O

=+ +

2 1

1 30.5

2

2 21

Kinetic model

1. N2O + * ↔ N2O*2. N2O* → N2 + O*3. 2 O* ↔ 2* + O2


N2O decomposition over Mn2O3Vannice et al. 1996

0.0 2.0 4.0 6.0 8.0 10.0

pO2 / kPa

0.0

0.1

0.2

0.3

0.4

r / 1

0-6m

ol.s

-1.g

-1

Oxygen inhibitionorder N2O ~0.78

Eaobs= 96 kJ/mol

648 K

638 K623 K608 K598 K

= Explain =

pN2O = 10 kPa


N2O decomposition over Mn2O3Vannice et al. 1995

Kinetic model

1. N2O + * ↔ N2O*2. N2O* → N2 + O*3. 2 O* ↔ 2* + O2

Rate expression

( )( )5.031

12

22

2

1 KppKpKNkr

OON

ONT

++=

Values

Ea2 130= kJ / mol

K J/mol 109SkJ/mol 92

3

3

=∆=∆H

K J/mol 38SkJ/mol 29

1

1

−=∆−=∆H

= Thermodynamically consistent =


Ethanol dehydrogenationFranckaerts &Froment

C2H5OH ↔ CH3CHO + H2

Model:

1. A + * ↔ A*2. A* + * ↔ R* + S* (r.d.s.)3. R* ↔ R + *4. S* ↔ S + *

= Derive rate expression =

Cu-Co cat.


Initial rates - linear transformation

Full expression

Initial rate

After rearrangement

{ }{ }

rk s N K p p p K

K p K p K pT A A R S eq

A A R R S S

=−

+ + +2

21

/

{ }r k K p

K pA A

A A0 21

=+

pr k K

Kk K

pA

A

A

AA

0

1= + ⋅

y a b x= + ⋅linear form:

linear least squares fittrends, positive parameters

Ethanol dehydrogenation


Kinetic studies

( ) ( )oii

iio

i

i

FWxrr

FWddx

⋅−=⇒⋅−=

νν

Rate equation

= model selection == rate parameters =

Differential reactor:• plug flow, low conversion• CSTR

Take care catalyst effectiveness = 1 !

Integral reactor


Model selection

Mechanistic informationInitial rate measurementsFit data on rate expressions (non-linear least squares)Best model:– low SSR (sum of squares of residuals)– no trending in residuals– parameters significant– parameters obey thermodynamics– ‘linearization’


Data fitting - Parameter estimation

Linear least squares - straightforwardNonlinear least squares methods:– Simplex / Powell etc. (iterative)– Levenberg-Marquardt (gradient)

( )O F Min x x

x f k K

obs calci

calc j

. .

(.... , ...)

= −

=

∑ 2

with:

to be estimated

= initial parameters values needed =


Model selection

Divergence rival modelsExperimental design (sequential)

po

rate

model 1

model 2

experimental range

Catalysis Engineering - Kinetics Catalysis Engineering - Kinetics

N2O decomposition over ZSM-5 (Co,Cu,Fe)

2 N2O 2N2 + O2

Kapteijn et al. 11th ICC,1996

Rate expression

( )r k N pk kT N O=

+1

1 2

2

1

Kinetic model

1. N2O + * → N2 + O*2. N2O + O*→ N2 + O2 + *

no oxygen inhibition


N2O decomposition over ZSM-5 (Co,Cu,Fe)Kapteijn et al. 11th ICC,1996

Rate expression

( )r k N pk k K p

T N O

O

=+ +

1

1 2 3

2

21

Oxygen inhibition model

1. N2O + * → N2 + O*2. N2O + O*→ N2 + O2 + *3. O2 + * ↔ *O2

0 2 4 6 8 10

p(O2) / kPa

0.0

0.2

0.4

0.6

0.8

1.0

X(N

2O)

Fe-ZSM-5 Co-ZSM-5

Cu-ZSM-5

743 K

833 K

793 K

733 K688

773 K


Effect of CO on N2O decomposition

0.0 0.5 1.0 1.5 2.0

molar CO/N2O ratio

0.0

0.2

0.4

0.6

0.8

1.0

X(N

2O)

Co-ZSM-5 (693 K)

Cu-ZSM-5 (673 K)

Fe-ZSM-5 (673 K)

CO removes oxygen from surfaceso ‘enhances’ step 2, oxygen removal

now observed: rate of step 1 r1 = k1 NT pN2O

increase: ~2, >3, >100

CO + O*→ CO2 + *

CO + * ↔ CO* (Cu+)


Effect of CO on N2O decomposition

rate without CO rate with CO

r k N pT N O= 1 2

So k1/k2 = : 1 Co>2 Cu>100 Fe

ratio = 1 + k1/k2 and:21

21* 1 kk

kkO +

=θ

θ O* = 0.7>0.9>0.99

( )21

1

12

kkpNkr ONT

+=


Apparent activation energies N2O decompositionCO/ N2O = 2

Cu ( )r k N pk k K p

k N pK p

T N O

CO CO

T N O

CO CO=

+ +≈1

1 2

12 2

1

Co, Fe

r k N pT N O= 1 2

E Eaobs

a= 1

E E Haobs

a CO= +1 ∆

Apparent activation energies (kJ/mol)

only N2O CO/N2O=2

Co 110 115

Cu 138 187

Fe 165 78


Apparent activation energies N2O decompositionCO/ N2O = 0

Co,Cu

Fe

( )r k N pk kT N O=

+1

1 2

2

1

r k N pT N O= 2 2 E Eaobs

a= 2

E E Eaobs

a a= mix( )1 2,

Apparent activation energies (kJ/mol)

only N2O CO/N2O=2

Co 110 115

Cu 138 187

Fe 165 78


Kinetic coupling between catalytic cyclesadsorption effect on selectivity

Hydrogenation: butyne -> butene -> butaneA1 A2 A3

butyne and butene compete for the same sitesbut: K1 >> K2resulting high selectivity for butene (desired) possibleeven when k2 > k1

since:22

112,1 Kk

KkS =

Meyer and Burwell (JACS 85(1963)2877) mol%:2-butyne 22.0cis-2-butene 77.2trans-2-butene 0.71-butene 0.0butane 0.1

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