1

MAE 5310: COMBUSTION FUNDAMENTALS

Introduction to Chemical Kinetics

September 24, 2012

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

2

CHEMICAL KINETICS OVERVIEW• In many combustion processes, chemical reaction rates control rate of combustion

• Chemical reaction rates determine pollutant formation and destruction

• Ignition and flame extinction are dependent on rate processes

• Overall reaction of a mole of fuel, F, with a moles of oxidizer, O, to form b moles of products, P, can be expressed by a global reaction mechanism as:

• From experimental measurements, rate at which the fuel is consumed expressed as:

• [i] is molar concentration of ith species in mixture

• Equation states that rate of disappearance of fuel is proportional to each of reactants raised to a power

• Constant of proportionality, kglobal, is called global rate coefficient, which is a strong function of temperature and minus sign indicates that fuel concentration decreases with time

• Exponents n and m relate to reaction order

– Reaction is nth order with respect to fuel

– Reaction is mth order with respect to oxidizer

– Reaction is (n+m)th order overall

bPaOF

mOn

FglobalF MMTk

dt

Md

3

EXAMPLE OF INTERMEDIATE SPECIES

• Consider global reaction of conversion of hydrogen and oxygen to water

• The following elementary reactions are important:

• First reaction produces hydroperoxy, HO2 and a hydrogen atom, H

– HO2 and H are called radicals

– Radicals, or free radicals, are reactive molecules, or atoms, that have unpaired electrons

• To have a complete picture of hydrogen and oxygen combustion over 20 elementary reactions are necessary

• Collection of elementary reactions necessary to describe an overall reaction is called a mechanism

MHOMOH

HOHHOH

OOHOH

HHOOH

OHOH

22

22

2

222

222 22

4

MOLECULAR KINETIC AND COLLISION THEORY OVERVIEW:BIMOLECULAR REACTIONS

• Molecular collision theory an be used to provide insight into form of bimolecular reaction rates and to suggest the temperature dependence of the bimolecular rate coefficient

• Consider a single molecule of diameter traveling at constant speed v and experiencing collisions with identical, but stationary, molecules

– If distance between traveled between collisions (mean free path, ) is large then moving molecule sweeps out a cylindrical volume in which collisions are possible = v2t in a time interval t.

– At ambient conditions for gases:

• Time between collisions ~ O(10-9 s)

• Duration of collisions ~ O(10-12 – 10-13 s)

– If stationary molecules distributed randomly and have a number density, n/V, the number of collisions experienced by the traveling molecule per unit time is: Z = collisions per unit time = (n/V)v2

• In actual gas all molecules are moving

– Assuming a Maxwellian distribution for all molecules, the collision frequency, Zc, is given by:

BAkdt

Ad

DCBA

rbimolecula

vV

nZc

22

5

MOLECULAR KINETIC AND COLLISION THEORY OVERVIEW:BIMOLECULAR REACTIONS

• So far theory applies to identical molecules

– Extend analysis to collisions between unlike molecules have diameters A and B. Diameter of collision volume is then given as AB=(A+ B)/2

– This is an expression for the frequency of collision of a single A molecule with all B molecules

• Ultimately we want collision frequency associated with all A and B molecules

– Total number of collisions per unit volume and per unit time is obtained by multiplying collision frequency of a single A molecule by the number of A molecules per unit volume and using the appropriate mean molecular speed (RMS)

– ZAB/V = Number of collisions between all A and all B / Unit volume Unit time

vV

nZ AB

Bc

22

BA

BA

bAB

BAc

BAABBA

c

mm

mm

Tk

V

n

V

nZ

vvV

n

V

nZ

21

2

21

222

8

6

MOLECULAR KINETIC AND COLLISION THEORY OVERVIEW:BIMOLECULAR REACTIONS

• NAvogadro = 6.022x1023 molecules/mol or 6.022x1026 molecules/kmol

• Probability, P, that a collision leads to reaction can be expressed as product of two terms

1. Energy factor, exp[-EA/RT]

• Expresses the fraction of collisions that occur with an energy above the threshold level necessary for reaction, EA, or activation energy

2. Geometrical or steric factor, p

• Takes into account the geometry of collisions between A and B

1 AvogadroAB PNV

Z

dt

Ad

RT

EATTk

RT

EATk

RT

ETkpNTk

BART

ETkpN

dt

Ad

An

A

AbABAvogadro

AbABAvogadro

exp

exp

exp8

exp8

21

2

21

2

More common curve fitA, n and EA are empirical parameters

7

EXAMPLE: H2 OXIDATION AND NET PRODUCTION RATES

021

22

22

2

222

222

0

,...,,

22

4

4

3

3

2

2

1

1

ii

nii

k

k

k

k

k

k

k

k

MM

tMtMtMfdt

tMd

MHOMOH

HOHHOH

OOHOH

HHOOH

OHOH

f

b

f

b

f

b

f

b

System of 1st order, ordinary differential equations

Initial conditions for each participating species

Global reaction

Partial mechanism

Find: d[O2]/dt, d[H]/dt, etc.

8

GENERAL NOTATION

jj

f

b

n

jjbii

n

jjfii

i

n

jjj

k

k

n

jjj

n

jjj

n

jjj

MkMkdt

Md

MM

MM

11

11

11

9

EXAMPLE

Determine the collision-theory steric factor for the reaction O + H2 → OH + H at T=2000 K give the sphere diameters, O=3.050 and H2=2.827 Å using the data in Appendix 2 of Glassman

Comments

• Pay attention to units:

– kb=1.381x10-23 J/K = 1.381x10-16 g cm2/s2 K