APPLIED REACTION KINETICS

Friday A01 09:00-11:00 (Assoc. prof. Bohumil Bernauer, A27, Bohumil.Bernauer@vscht.cz,

+420220444017 )

Friday BS9 13:00-15:00 (Dr. Milan Bernauer, H08, Milan.Bernauer@vscht.cz,

+420220444287)

κίνησις "kinesis", movement or to move

Resources and references

• Notes from lectures

• Internet

web.vscht.cz/bernauem/

• Textbooks

Fogler Scott H.: Elements of Chemical Reaction Engineering, 4th Edition, Prentice

Hall, 2006. (http://www.engin.umich.edu/~cre/)

Missen R.W., Mims C.A., Saville B.A., Introduction to Chemical Reaction

Engineering and Kinetics, J. Wiley&Sons, N.Y. 1999.

• Journals (on-line)

• Software

MS Excel, (Matlab, Octave, Athena Visual Studio, FORTRAN,

Maple….)

Chemical reactor(s) heart of

chemical process

Raw material separation reaction separation product

Fischer-Tropsch (SASOL, RSA ) WGS (BASF, FRG)

Methane aromatization (ICTP, CZ)

N2O decomposition (IPC AS, CZ)

( , , , ..., )r composition T P params

Summary of the 1st lecture • Stoichiometry • Extent of reaction • Fractional conversion of key component • Stoichiometric matrix • Balance of chemical elements • Selectivity, Yield • Reaction rate definition

2NO N2 + O2 closed (batch) system

Stoichiometry

t =0 t > 0

Atoms of oxygen

Atoms of nitrogen

2

2o o

NO On n

2

2NO On n

2

2o o

NO Nn n

2

2NO Nn n

2 2

2 2o o

NO O NO On n n n

2 2

2 2o o

NO N NO Nn n n n

2 2

2( )o o

NO NO O On n n n

2 2

2( )o o

NO NO N Nn n n n

2 2 2 2

( ) ( )

2 1 1

o oo

O O N NNO NOn n n nn n

-2NO + O2 + N2 = 0 Symbols for species NO = A1 O2 = A2 N2 = A3

Stoichiometric coefficients 1 = -2 2 = 1 3 = 1

3

1

0 produ cts

0 0 inerts

0 reactants

i

i i i

i

i

A

" " the m aterial elem ent (P lato)

" " the count, the quantity

Molar extent of the reaction [ksi:]

2 2 2 2

( ) ( )

2 1 1

o oo

O O N NNO N

i

O

o

i

i

n

n n n nn n

n

From the definition of the reaction extent follows:

1. The reaction extent has the dimension of moles (number of molecules)

2. The reaction extent value depends on stoichiometry of reaction

3. The reaction extent is an extensive variable

t = 0 t > 0

Reaction extent for a single reaction in closed (batch) system

ino

in

o

i i

i

n n

o

i i in n

Closed system

1

0 produ cts

0 0 inerts

0 reactan

ts

i

N

i i i

i

i

A

Example

2NO N2 + O2

1 2 2 3 2

1 2 1

1

2

3

1

2

3

N O , N , O

2, 1, 1

2 1 1 ,

2 1 1 0

T

T

A A A

A

A

A

A

A

A

ν A

ν A

Matrix notation

Fractional conversion of key component, j

m ax

jX

j

o

jjnn

*

max

o

j

i

o

i

i

j

j

o

j

i

o

ii

jn

nn

n

nn

X

max

jX

max

o

j

j j

j

nX X

j

o

j

j

io

iiXnnn

(0,1)

o

j j

j jo

j

n nX X

n

100 (0,100)

o

j j

j jo

j

n nX X

n

Number of moles of key

component in limits

(chemical equilibrium or 0)

*0

jn

i j

Stoichiometric matrix in the case of several reactions

1

0 1,

N

ki i

i

A k NR

11 12 1 1

21 22 2 2

,1 ,2 ,

...

...,

. .

...

N

N

NR NR NR N N

A

A

A

ν A

Stoechimetric matrix has NR rows and N columns

reaction component

νA 0

1

NR

o

i i ki k

k

n n

o

ki ki

k

ki

n n

Number of moles of i-th component consumed or created in k-th reaction :

o T n n ν ξ

1 11

2 22, ,

.. ....

o

o

o

o

N NRN

n n

n n

n n

n n ξ

o

ki kin n

Molar extent of k-th reaction :

Number of moles of i-th

components:

Matrix notation

Problem 1.1

Oxidation of ammonia on Pt-Rh catalyst

NH3 + 1.25 O2 NO + 1.5H2O (1)

NH3 + O2 0.5 N2O + 1.5H2O (2)

NH3 + 0.75O2 0.5 N2 + 1.5H2O (3)

Task: To write down the stoichiometric matrix.

A1 A2 A3 A4 A5 A6

Reaction NH3 O2 NO N2O N2 H2O

(1) -1 -1.25 1 0 0 1.5

(2) -1 -1 0 0.5 0 1.5

(3) -1 -0.75 0 0 0.5 1.5

Molar balance table of component in closed (batch) system

Component t = 0 t > 0

NH3 1

on

1 1 1 2 3( )

on n

O2 2

on

2 2 1 2 31.25 0.75

on n

NO 3

on

3 3 1

on n

N2O 4

on

4 4 20.5

on n

N2 5

on

5 5 30.5

on n

H2O 6

on

6 6 1 2 31.5( )

on n

6

1

o

i

i

n

6

1 3

1

0.25o

i

i

n

3

1 1 2 3

1 6

1 3

1

e.g . m olar fraction of N H

( )

0 .25

o

o

i

i

ny

n

Independent reactions

Set of NR reactions in reaction network is independent if

Rank()=NR

or

number of independent reactions = Rank()

Problem 1.2

NH3 + 1.25 O2 NO + 1.5H2O (1)

NH3 + O2 0.5 N2O + 1.5H2O (2)

NH3 + 0.75O2 0.5 N2 + 1.5H2O (3)

2NO N2 + O2 (4)

Task: To calculate the number of independent reactions.

We determine the rank of stoichiometric matrix by Gaussian elimination:

1 1.25 1 0 0 1.5 1 1.25 1 0 0 1.5

1 1 0 0.5 0 1.5 0 0.25 1 0.5 0 0

1 0.75 0 0 0.5 1.5 0 0.5 1 0 0.5 0

0 1 2 0 1 0 0 1 2 0 1 0

1 1.25 1 0 0 1.5 1 1.25 1 0 0 1.5

0 0.25 1 0.5 0 0 0 0.25 1 0.5 0 0

0 0 1 1 0.5 0 0 0 1

0 0 2 2 1 0

1 0.5 0

0 0 0 0 0 0

Rank()=3 only 3 reactions are independent

Extent of the reaction in a flow system

1

0 1,

N

ki i

i

A k NR

o

iF

Open (flow) system in a steady state

iF

1

, -outlet, inlet m olar flow rates of i-th species [m ole/s]

extent of k-th reaction per unit of tim e [m ole/s]

m ean residence tim e [s]

NR

o k

i i ki k k

k

o

i i

k

T

F F

F F

oF = F + ν ξ

& &

&

&1 1 1

2 2 2, ,

.. .. ..

o

o

o

o

N N NR

F F

F F

F F

F F ξ

&

&&

&

Inlet molar flow rates

[mole of i-th species/s]

Outlet molar flow rates

[mole of i-th species/s]

REACTOR

A reaction is at steady-state if the concentration of all species in each

element of the reaction space (i.e. volume in the case of homogeneous

reaction or surface of catalyst in the case of catalytic heterogeneous

reaction) does not change in time.

Balance of chemical elements

1

0 1,

N

ki i

i

A k NR

Open (flow) system at steady state

, -outlet, inlet m olar flow s

of j-th chem ical elem ent [m ole/s]

o

o

j

j j

j

oΦ = Φ

1 1

2 2,

.. ..

o

o

o

NEL NEL

oΦ Φ

o

j

j

REACTOR

NEL - Number of

chemical elements

Formula matrix E

N components

N=6

N H O

NH3 1 3 0

O2 0 0 2

NO 1 0 1

N2O 2 0 1

N2 2 0 0

H2O 0 2 1

NEL elements NEL=3

There are no creation or annihilation of chemical elements in chemical reactions:

νE = 0

11 12 1 11 12 1,

21 22 2 21 2 ,

,1 ,2 , ,1 ,2 ,

... ... 0 0 ... 0

... ... ... 0 0 ... 0

. . .

... ... 0 0 ... 0

N NEL

N NEL

NR NR NR N N N N NEL

E E E

E E

E E E

NR

NEL

Formula vector of NH3

Molar* weight (relative molecular mass) of i-th species:

1

1 1

2 2,

.. ..

NEL

i ij j

j

N NEL

M E m

M m

M m

M m

M = Em

M m

and we have for molar weights of species

0

because 0

νM = νEm

νE

Atomic weights (relative atomic mass) of j-th element

*The mole is the amount of substance of a system which contains as many elementary

entities as there are atoms in 0.012 kilogram of carbon 12.

Avogadro constant = 6.022 141 29(27) × 1023 mol−1 (http://www.nist.gov)

Balances of atoms in batch and flow systems

C losed (batch) system :

O pen (flow ) system at steady state:

because

T

TT T T T T T

T

TT T T T T T

o

o o o

o

o o o

n = n + ν ξ

E n = E n + E ν ξ E n + νE ξ E n

F = F + ν ξ

E F = E F + E ν ξ E F + νE ξ E F

νE 0

&

& &

Finally

T T

T T

o o

o o

E n n E n n 0

E F F = E F F 0

These equations are used

in data reconciliation tasks

around chemical reactors.

The last equation (flow

system) is valid only at

steady state.

Problem 1.3

Selective reduction of NOx by C3H8

NO NO2 CO C3H8 CO2 H2O O2 N2

4.563 0.1845 0 2.943 0 0 90 0

2.2905 2.3355 0 2.898 x x x x

[ / min]o

iF mole

o

iF i

F

[ / m in]iF mole

Calculate the missing outlet molar flow rates

Steady state,

unknown

stoichiometry

of reactions

T o T E F - F = E F 0

1-NO 2-NO2 3-CO 4-C3H8 5-CO2 6-H2O 7-O2 8-N2

N 1 1 0 0 0 0 0 2

O 1 2 1 0 2 1 2 0

C 0 0 1 3 1 0 0 0

H 0 0 0 8 0 2 0 0

1

2

5

6

3

4

7

8

F

F

F

F

F

F

F

F

unknown

iF

1

TE 2

TE

1 2

2 1

1 1

2 1 2 1

T T

KNOWN UNKNOWN

T T

UNKNOWN KNOWN

T T T T

UNKNOWN KNOWN KNOWN

E F E F 0

E F E F

F E E F Q F Q E E

known

iF

0 0 1 3

0 0 0 4

0.5 1 0.5 5

0.5 0.5 0 0

Q

We obtain using Excel (homework 1)

NO NO2 CO C3H8 CO2 H2O O2 N2

4.563 0.1845 0 2.943 0 0 90 0

2.2905 2.3355 0 2.898 0.135 0.18 88.76 0.061

and taking we have 1

2 2 1 1 1

T T

F E E F Q F

[ / min]o

iF mole

[ / m in]iF mole

Selectivity Moles of a particular product generated per mole of key reactant consumed

Yield Moles of a particular product generated per one initial mole of key reactant

Component t = 0 t > 0

NH3 1

on

1 1 1 2 3( )

on n

O2 2

on

2 2 1 2 31.25 0.75

on n

NO 3

on

3 3 1

on n

N2O 4

on

4 4 20.5

on n

N2 5

on

5 5 30.5

on n

H2O 6

on

6 6 1 2 31.5( )

on n

Ammonia oxidation

3

3

3

3 1 1

1 2 3 1 2 3

0

3 1 1

1 1

( ) ( )

o

o

NO NH

no

NO NH o o

nS

nY

n n

Reaction rate

(IUPAC Gold Book = rate of conversion )

11 m ole.s

i

i

k

k

dndr

dt dt

dr

dt

one reaction

several reactions

1 1

. .

. .

i

i

i

A i i

NR NR

Tk

A ki ki k

k k

dn dR r r

dt dt

dR r

dt

R = ν

R = ν r

Rate of generation (consumption) of component Ai

Closed system

of uniform

pressure,

temperature

and

composition.

t

m ax

0; 0

d

d t

exp exp exp O bjective function

m inim ization of = least squares solution of overdefined system (N >N R )

TT T T

R ν r r R ν r R ν r

r

We measure usually the rates of generation (consumption) of

components R and we want to calculate r

Problem 1.4

Steam reforming of methane (5 species, N=5, 2 reactions, NR=2)

1 2 3 4

CH4 + H2O CO + 3H2 (1)

5

CO + H2O CO2 + H2 (2)

Measured R : (-0.97572, -2.88778, -1.02127, 4.832804, 2.078537)T (mol/s)

11 1 1 3 0 12 3

0 1 1 1 1 3 4

T T

ν νν νν ν-0.10256 -0.02564 0.179487 0.230769 -0.07692

0.076923 -0.23077 -0.38462 0.076923 0.307692

1

2

0.94619

1.99546

r

r

1

expT

r νν νR

Specific reaction rate

The reaction rate is, like , an extensive property of the system, a specific rate (intensive property) is obtained by dividing by the total volume, mass, surface of the system:

3 1

,

1 1 1 1 m ole.m .

1

R eaction rate per vo um

1

l e

i

V

i

k

V k k

dndr r s

V V dt V dt

dr r

V V dt

1 1

,

1 1 1 1 m ole.kg .

1 1

R eaction rate per m ass

i

M

i

k

M k k

dndr r s

m m dt m dt

dr r

m m dt

d

d t

d

d t

2 1

,

1 1 1 1 m ole.m .

1

R eaction rate per surface

1

i

S

i

k

S k k

dndr r s

S S dt S dt

dr r

S S dt

1

,

R eaction rate per active center (turn

1 1 1 1 s

1

over num be

1

r)

i

RS

RS RS RS i

k

RS k k

RS RS

dndr r

n n dt n dt

dr r

n n dt

1 2, ,

C entral

, .

proble

.. , , catalytic activity, transport param e ters,...

m of APPLIED C H EM IC AL K IN ET IC

.

S

Nr function T c c c P

Rule 1: The rate function r at constant temperature generally

decreases in monotonic fashion with time (or extent or conversion).

Rule 2: The rate of irreversible reaction can be written as

1 2( ) , , ...

Nr k T g c c c

Rule 3: The rate constant k depends on temperature (Svante

Arrhenius, 1889):

R( )

E

Tk T Ae

Rule 4: The function g is independent of temperature:

1 2

1 2 1 2

1

, , ... ... N i

N

N N i

i

g c c c c c c c

Rule 5: When a reaction is reversible:

1 2 1 2( ) , , ... ( ) , , ...

f b f f N b b Nr r r k T g c c c k T g c c c

Problem 1.5 (homework 2)

In flow catalytic reactor the synthesis of methanol is carried out

CO(g) + 2H2(g) CH3OH(g)

The inlet mass flow rate of CO is 1000 kg.h-1 of CO and the inlet flow rate of

hydrogen is supplied so that the inlet molar ratio H2:CO is equal to 2:1. 1200 kg of

the catalyst is placed in the reactor.

The outlet mass flow of CO is 860 kg.h-1.

To determine:

1. Mean reaction rate per mass of catalyst in mol.kg-1.s-1 .

2. If specific internal surface of catalyst is 55 m2.g-1, calculate mean reaction rate

per surface of catalyst in mol.m-2.s-1.

3. If per 1 m2 of catalyst contains 1019 active sites, calculate mean reaction rate

per active site in s-1.

4. Calculate inlet and outlet gas mixture composition in molar fractions.

Data:

MCO= 28.010 kg.kmol-1

NA=6.0221413x1023 mol-1 (Avogadro number)

1 2 2 3 3

1

, ,

1 1

2 , 2

1 1

, ,

NR

o o

i i ki k i i i

k

o o o

i i i i i i

M S RS

CAT i CAT i RS i RS

CO A H A CH OH A

r

F F F F

F F F F F Fr r r

m m S S n n

ν R

& &

& & &

inlet outlet

1-CO

2-H2

3-CH3OH

1

oF

12

oF

0

13

o oF F

o

iy

1

1

1

1

3 3

o

o

o

Fy

F

1

2

1

2 2

3 3

o

o

o

Fy

F

0

1 11

oF F &

2 22

oF F &

3 31

oF F &

1 2o

F F &

iy

1

1

1

1

3 2

o

o

Fy

F

&

&

1

2

1

2 2

3 2

o

o

Fy

F

&

&

3

13 2

oy

F

&

&

1

1 1 11

oF F X

1

1 1 1

1 1

Fractional conversion of key com ponent

1 key com ponent

/

o o

o oi i i

i i

MAX i

F F FX F F F X

&

&

2 1 1 12 2

o oF F F X

3 1 1

oF F X

1 1 13 2

o oF F F X

1

3 1 1 8 2 1

3

23

3 1

3 19

860 1000 / 0.02801 / 36001, 1.38839 m ol.s

1

1.38839 1.388391.157 10 m ol.kg . , 2 .104 10 m ol.m .

1200 55 10 1200

1.38839 6.0221413 101.267 10 s

55 10 1200 10

M S

RS

i

r s r s

r

&

inlet outlet

1-CO

2-H2

3-CH3OH

o

iy

1

1

1

1

3 3

o

o

o

Fy

F

1

2

1

2 2

3 3

o

o

o

Fy

F

0

1

1 1 1

1

1 1 1 1

1 10.3162

3 2 3 2

o

o o

F X Xy

F F X X

1

1000 8600.14

1000X

11 1 1

2

1 1 1 1

2 12 20.6324

3 2 3 2

o o

o o

XF F Xy

F F X X

iy

1 1 1

3

1 1 1 1

0.051473 2 3 2

o

o o

F X Xy

F F X X

1