Chemical Kinetics
§ Chemical kinetics deals with the study of elementary
reactions, determination of their rates, and development of
reaction mechanisms. It is a specialized field of physical
chemistry.
§  On the theoretical side, it involves fundamental analysis based on
reaction rate theories and semi-empirical approaches, as well as more
advanced theories based on quantum chemisty, for developing the
appropriate reaction mechanisms. On the experimental side, it deals with
the determination of reaction rate constants and the development of
mechanisms using experimental data.
§  During the last four decades, there has been remarkable progress in this
field, as reasonably reliable and detailed mechanisms have been
developed for H2, CO, and several hydrocarbon and renewable fuels for
a range of conditions in terms of pressure, equivalence ratio, etc.
§  It remains an active area of research, especially due to our interest in
developing advanced predictive capabilities for reacting flows, and kinetic
models for both conventional and renewable fuels.
Chemical Kinetics
§  Reacting flows are characterized by several time scales, for example,
reaction time, convection or flow time, diffusion time.
§  The ratio of transport time to reaction time is defined as the Damkohler
number (Da), one of the most important parameters in the study of
combustion.
§  For Da << 1 (chemically frozen limit); the effect of kinetics may be
neglected in some flow regime or domain.
§  Da >> 1 (equilibrium limit), the effect of finite rate chemistry may be
neglected or circumvented in the numerical/ theoretical analysis. This
leads to significant simplification in the analysis, for example, diffusion
flames that are far from extinction.
§  However, the chemical time and transport time are often of the same
order of magnitude, and the inclusion of finite rate chemistry is
essential in the analysis of reacting flows.
€
Law of Mass Action
Law of mass action provides a rate expression for any given reaction. Any elementary
reaction can be written in a general form as:
NS
NS
∑υ #M = ∑υ ##M
i
i
i=1
i
(1)
i
i=1
Here ν is the stoichiometry coefficient, and M a symbol for chemical species. The
subscript i=1, 2,..Ns, with Ns being the total number of species involved in the reaction. It
is important to distinguish between elementary and global reactions. Reactions 2a and
2b €
are elementary reactions, since they occur at the molecular level, while reaction (3) is
a global reaction as it does not occur at the molecular level. Chemical kinetics deals with
determining rates of both types of reactions.
O2 + H ⇒ OH + O
(2a)
O + H 2 ⇒ OH + H
H 2 + 0.5O2 ⇒ H 2O
(2b)
(3)
For elementary reactions represented by Eq. (1) or (2), the mass conservation yields
dN1 dN 2
dN i
dξ
##
#
=
....... =
= (υ i − υ i )
dt
dt
dt
dt
€
(4)
€
Phenomenological Law of Mass Action
Here Ni is the number of moles of species i, and ξ is the reaction progress variable
indicating the degree of advancement of reaction. Dividing by the system volume, we
can write the above equation in terms of the rate of change of concentration of each
species as
dc1 dc 2
dc i
dξ 1
$$
$
˙
=
....... =
= ω i = (υ i − υ i )
= (υ i$$ − υ i$ )ω
dt
dt
dt
dt V
(5)
Here ω is the reaction rate in moles/volume/time. Based on experimental and
theoretical studies, the phenomenological law of mass rate expression has been
developed, which states that the reaction rate (ω) is proportional to the product of the
concentrations of reactants.
ω = k.Π Ni s c υi $i = k.c1υ 1$ c υ2 $2 ...
(6)
k is defined as the reaction rate constant, and is
generally expressed as
k = AT α e−(E / R u T )
(7)
Here A is the pre-exponential factor, E the activation energy, Ru the universal gas
constant, and T the temperature (K). Equation 7 is known as the Arrhenius equation.
Thus for reaction (2a), one can write
d[ H ]
d [O2 ] d [OH ]
−
=−
=
= ω = k [O2 ][ H ]
dt
dt
dt
Multiple Reactions
Fuel oxidation chemistry generally involves a number of elementary reactions,
€
which can be expressed by rewriting the equation (1) for the jth reaction as
NS
∑υ
NS
i, j
# M = ∑υ ## M
i
i, j
i
i=1
(8)
i=1
Here j=1,2,..NR, with NR representing the total number of reactions. Then the
rate of consumption (or production) of species i (moles/vol/time) is
€
ωi =
§  Thus
dc i
=
dt
NR
∑ (υ i, j $$ − υ i, j $ )ω j
with
NS
i
ω j = k j .Π c
υ $i , j
i
(9)
j=1
the rate of consumption (or production) of any species can be determined if
the reaction mechanism and the reaction rate constant for each reaction in the
mechanism are known.
€ terms in the species and energy equations,
§  These equations represent the source
which will be discussed later.
Forward and Backward Reactions
In a reacting system, reactions (for example, reaction (2)) can proceed in
both forward and backward directions. Then equation (8) should be written as
NS
NS
∑υ
i, j
# M ⇔ ∑υ ## M
i
i, j
i
i=1
where j=1,2,..NR/2
(10)
i=1
Then the net forward reaction rate for the jth reaction is
€
υ$
υ $$
ω j = k j, f .Π Ni S c i i, j − k j,b .Π Ni S c i i, j
(11)
and the consumption or production rate of species (i) is
NR /2
€
ωi =
∑ (υ
j=1
€
i, j
$$ − υ $ )ω
i, j
j
(12)
Reaction Mechanism for H2-O2
1. h+o2=o+oh 2. o+h2=h+oh 3. oh+h2=h+h2o 4. o+h2o=oh+oh 5. h2+m=h+h+m 6. o2+m=o+o+m 7. oh+m=o+h+m 8. h2o+m=h+oh+m 9. h+o2(+m)=ho2(+m) 10. ho2+h=h2+o2 11. ho2+h=oh+oh 12. ho2+o=oh+o2 13. ho2+oh=h2o+o2 14. h2o2+o2=ho2+ho2 15. h2o2+o2=ho2+ho2 16. h2o2(+m)=oh+oh(+m) 17. h2o2+h=h2o+oh €
18. h2o2+h=h2+ho2 19. h2o2+o=oh+ho2 20. h2o2+oh=h2o+ho2 €
21. h2o2+oh=h2o+ho2 Reaction, NR=21
Species: Ns=7
H, O2, O, OH, H2, H2O, H2O2
NS
NS
∑υ
i, j
# M = ∑υ ## M
i
i, j
i
i=1
i=1
j=1, 2, … NR
dc
€
ωi = i =
dt
NR
∑ (υ
i, j
$$ − υ $ )ω
i, j
j
j=1
NS
i
ω j = k j, f .Π c
υ $i, j
i
NS
i
− k j,b .Π c
For practice, write equations
for the rate of reaction # 3 and
5, and for the net rate of
production of H2O
υ $i,$ j
i
Collision Theory of Reaction Rates
Since reactions at the molecular level involve collisions between molecules, the
reaction rate for a bimolecular reaction should be proportional to the collision
frequency multiplied by the probability that a collision leads to reaction. The
probability implies that only those molecules that possess energy greater than a
certain threshold value will react, and is proportional to exp(-E/ RuT).
The collision frequency is determined using a simplified model based on ideal gas
assumption. As depicted in the figure, the model considers two reactant species
(say) A and B. It assumes that colliding molecules only possess translational
energy. Let us consider the motion of one molecule (species A). It will collide with all
B molecules that are contained in the cylindrical volume (V) swept by this molecule.
Then the number of collisions encountered by this molecule is
2
Z A = Vn B = πσ AB
U AB n B
Here nA is the number density of A molecules, σAB (cylinder radius) =0.5.(σA+σB), and
UAB is the average velocity of A molecules with respect to B molecules.
€
Collision Theory of Reaction Rates
σB
σA
σA + σB
UAB
Considering nA as the number density of A molecules, the collision frequency
can be written as
2
2
Z = πσ AB
U AB n B n A = πσ AB
U AB CB CA Av2
(1)
Collision Theory of Reaction Rates
Then the reaction rate can be written as
2
ω˙ = (Z / Av )e−(E / R u T ) = πσ AB
U AB CB CA Av e−(E / R u T )P
(moles/m3/s)
(2)
Here exponent factor represents the probability of collision having sufficient
energy (or proportion of collisions with energy greater that E) for reaction, and P
is defined as the steric factor that accounts for the geometry of collisions, i.e.,
colliding molecules require certain orientation for reaction. UAB is given by
(assuming Maxwell velocity distribution)
U
2
AB
8k b T
mA mB
=
with µAB =
πµAB
mA + mB
(3)
Then the reaction rate can be written as
€
2
ω˙ = σ AB
(π 8kb T / µAB )1/ 2 CB CA Av e−(E / R u T )P
(4)
Comparing this expression for the reaction rate with that given by the law of
mass action yields an expression for the reaction rate constant as
€
Collision Theory of Reaction Rates
2
k(T) = σ AB
(π 8k b T / µAB )1/ 2 Av e−( E / R u T )P
€
(5)
P and E (activation energy) are determined from more advanced theories. Note
as discussed earlier, based on the law of mass action, the reaction rate is also
be expressed as
ω˙ = AT α CB CA e−(E / R u T )
(6)
Then the reaction rate constant can be written as
2
k(T) = AT α e−( E / R u T ) = σ AB
(π 8k b T / µAB )1/ 2 Av e−( E / R u T )P
€
(7)
This yields an expression for the preexponnetial factor (A) in terms of the
molecular properties. Also it gives α=0.5, which may not be correct for many
reactions.
Equilibrium Constants
Since any reaction can proceed in both forward and backward directions,
reaction (2) can be rewritten as
O2 + H ⇔ OH + O
(13)
Then the net forward rate for reaction (13) is
€
ω˙ = k f Π Ni S c υi $i − k b Π Ni S c υi $i$
(14)
If the reaction is in equilibrium, then the equilibrium constant for this reaction can be
defined in terms of concentrations as
kf
Kc =
= Π Ni S c (i,eυ #i#−υ #i )
kb
€
€
(15)
Since equilibrium constant can be determined experimentally and with better
accuracy, it is commonly used to calculate the backward reaction rate constant. Thus
€
ω˙ = k f (Π Ni S c υi $i − K c−1Π Ni S c υi $i$ )
(14a)
In some cases, the backward reaction rate is much slower than the forward
rate, especially when the product species are radical species (see Eq. 13),
whose concentrations are usually small.
Various Equilibrium Constants
Kc can be related to the other two equilibrium constants, defined in terms of
mole fractions and partial pressures, as
o
K c = K p ( p /RuT)
∑ υ "i"−υ "i
K c = K x ( p /RuT)
∑ υ "i"−υ "i
(16)
Here Kx and Kp (recall the discussion of equilibrium constant based on
species partial pressures) are defined as
€
NS
i
Kx = Π x
(υ #i# −υ #i )
i,e
€
K p = Π Ni S ( pi / p o )(υ #i#−υ #i )
(17)
In writing the above equations, we have used pi=ciRuT and xi = pi/p. Further Kx and Kp
are related through
€
€
o ∑ υ "i" −υ "i
K p = Kx ( p / p )
(18)
Use above relations for reaction (13) and for
O2 ⇔ 2O
€
CO + H 2O ⇔ CO2 + H 2
(19)
Steady State Approximation and Partial Equilibrium
§  In a reacting flows, equations (9) provide the source (sink) terms in the
conservation equations for species and energy. Thus if the reaction
mechanism and reaction rate constants are known, it provides a closed
system for direct numerical simulations (DNS) of chemically reacting flows.
§  However, development of a comprehensively validated mechanism along
with the reaction rate constants has been one of the major challenges in
simulating reacting flows. Even if a reliable mechanism is available, DNS of
reacting flows has been an extremely challenging problem due to (i) the
size of the mechanism, and (ii) a wide range of relevant length and time
scales that are needed to be resolved.
§  Consequently, various approximations or assumptions are invoked to
obtain an approximate but realistic solution. Using these approximations,
such as steady state and partial equilibrium assumptions, a detailed
mechanism can be simplified to obtain a reduced mechanism.
§  Depending upon the system complexity, the fuel oxidation chemistry may
be represented by a relatively small number of reactions, i.e., a skeletal or
semi-global mechanism. In the extreme case, the chemistry may be
represented by a single reaction.
Steady State Approximation
Fuel oxidation chemistry generally involves number of reactions and species.
Many of these species are radical species which play the role of intermediates
or chain carriers. While some of these radical species are continuously being
produced and consumed at rapid rates, their concentrations as well as their
net rate of change of concentration is small. For example,
H 2 + OH ⇔ H 2O + H
The net rate of change of H concentration is small compared to its rate of
production and consumption
€
dc i
= ω˙ i << k f Π Ni S c υi $i or k b Π Ni S c υi $i$
dt
N S υ #i
N S υ #i#
k
Π
c
=
k
Π
Then
f
i
i
b
i ci
(20)
(21)
This reduces the differential equation to an algebraic equation for the
€ solution of ci and thus reducing the overall computational effort.
Note: Above
approximation does not mean that the concentration of ci is not
€
changing, rather its rate of change is small relative to its production and
consumption rates.
Partial Equilibrium Assumption
The assumption implies that in a multi-reaction scheme, the net reaction rate
of (say) reaction j is small compared to its forward and backward reaction
rates. Thus
υ $i , j
ω j << k j, f .Π Ni S c i
NS
i
k j, f .Π c
€
υ #i, j
i
υ $i$, j
or k j,b .Π Ni S c i
NS
i
≈ k j,b .Π c
υ #i,# j
i
(22a)
(22b)
§  Note the difference between this and the the steady state approximation,
€
as the latter assumes that the rate of change of a particular species
concentration is small.
§  It reduces the number of reactions in the mechanism, and provides an
equation for the concentration of a particular species to be solved in
terms of other species concentrations.
€
Chemical Kinetics
Elementary Reactions versus Global Reactions
O2 + H ⇔ OH + O
Elementary Reaction
2H 2 + O2 ⇒ 2H 2O
Global Reaction
€
Global Reaction Rate and Reaction Order
a
−(E / R u T )
€ ω˙ = Ae
[ H 2 ] [O2 ]
b
A, E, a and b are determined through
curve fitting with experimental data
Types of Reactions
§  Unimolecular
H2 ⇔ H + H
H + O2 ⇔ OH + O
H + O2 + M ⇔ HO2 + M
§  Bimolecular
§  Termolecular
€