Page 1
14
Chemical Kinetics
14.1 Introduction
Up until now it has been assumed that chemical reactions take place very rapidly, and that
the equilibrium conditions are reached instantaneously. While most combustion processes
are extremely fast, often their speed is not such that combustion can be considered to be
instantaneous, compared with the physical processes surrounding them. For example, the
combustion in a reciprocating engine running at 6000 rev/min has to be completed in
about 5 ms if the engine is to be efficient. Likewise, combustion in the combustion
chamber of a gas turbine has to be rapid enough to be completed before the gas leaves the
chamber. Such short times mean that it is not possible for all the gases in the combustion
chamber to achieve equilibrium - they will be governed by the chemical kinetics of the
reactions.
Chemical kinetics plays a major role in the formation of pollutants from combustion
processes. For example, oxygen and nitrogen will coexist in a stable state at atmospheric
conditions, and the level of oxides of nitrogen (NO,) will be negligible. However, if the
oxygen and nitrogen are involved in a combustion process then they will join together at
the high temperature to form NO, which might well be frozen into the products as the
temperature drops. This NO, is a pollutant which is limited by legislation because of its
initant effects. NO, is formed in all combustion processes, including boilers, gas turbines,
diesel and petrol engines; it can be removed in some cases by the use of catalytic
converters.
It was shown in Chapters 12 and 13 that significant dissociation of the normal products
of combustion, carbon dioxide and water can occur at high temperature. The values
shown were based on the equilibrium amounts of the substances, and would only be
achieved after infinite time; however, the rates at which chemical reactions occur are
usually fast and hence some reactions get close to equilibrium even in the short time the
gases are in the combustion zone. An analysis of the kinetics of reactions will now be
presented.
14.2 Reaction rates
Reaction rates are governed by the movement and breakdown of the atoms or molecules in
the gas mixture: reactions will occur if the participating ‘particles’ collide. The number of
Page 2
14
Chemical Kinetics
14.1 Introduction
Up until now it has been assumed that chemical reactions take place very rapidly, and that
the equilibrium conditions are reached instantaneously. While most combustion processes
are extremely fast, often their speed is not such that combustion can be considered to be
instantaneous, compared with the physical processes surrounding them. For example, the
combustion in a reciprocating engine running at 6000 rev/min has to be completed in
about 5 ms if the engine is to be efficient. Likewise, combustion in the combustion
chamber of a gas turbine has to be rapid enough to be completed before the gas leaves the
chamber. Such short times mean that it is not possible for all the gases in the combustion
chamber to achieve equilibrium - they will be governed by the chemical kinetics of the
reactions.
Chemical kinetics plays a major role in the formation of pollutants from combustion
processes. For example, oxygen and nitrogen will coexist in a stable state at atmospheric
conditions, and the level of oxides of nitrogen (NO,) will be negligible. However, if the
oxygen and nitrogen are involved in a combustion process then they will join together at
the high temperature to form NO, which might well be frozen into the products as the
temperature drops. This NO, is a pollutant which is limited by legislation because of its
initant effects. NO, is formed in all combustion processes, including boilers, gas turbines,
diesel and petrol engines; it can be removed in some cases by the use of catalytic
converters.
It was shown in Chapters 12 and 13 that significant dissociation of the normal products
of combustion, carbon dioxide and water can occur at high temperature. The values
shown were based on the equilibrium amounts of the substances, and would only be
achieved after infinite time; however, the rates at which chemical reactions occur are
usually fast and hence some reactions get close to equilibrium even in the short time the
gases are in the combustion zone. An analysis of the kinetics of reactions will now be
presented.
14.2 Reaction rates
Reaction rates are governed by the movement and breakdown of the atoms or molecules in
the gas mixture: reactions will occur if the participating ‘particles’ collide. The number of
Reaction rates 277
collisions occurring in a mixture will be closely related to the number densities (number
per unit volume) of the 'particles'. The number density can be defined by the molar
concentration, c, which is the amount of substance per unit volume. This is obviously a
measure of the number of particles per unit volume since each amount of substance is
proportional to the number of molecules; this is illustrated in Fig 14.1. The molar
concentration will be denoted by enclosing the reactant or product symbol in [ 1.
.. .
0.
.
. 0.
.O .
.
..
1: 'd.1
Fig. 14.1 Diagrammatic representation of molar concentration. Molar concentration in (a) is
approximately half that in (b)
Reactions occur when two, or more, reactants are capable of reacting. Many simple
chemical reactions are second order, e.g.
kf
A+B-C+D
kb
(14.1)
kf
C02 + H2-H20 + CO
kb
where the first reaction is a general one and the second is an example based on the water
gas reaction (chosen because the same number of reactants and products exist on both
sides).
These reactions can be written as
(14.2)
where A,, A,, .. . etc are the elements or compounds involved in the reaction, and
Y,, v2, . . . etc are the respective stoichiometric coefficients. Equation (14.2) can be further
generalised to
(14.3)
where q = total number of species being considered, v' represents the stoichiometric
coefficients of the reactants, and Y" that of the products.
In this case q has been taken as the same on both sides of the equation because it is
assumed that the same species can exist on both sides. If a species does not exist on one of
the sides it is represented by a stoichiometric coefficient of zero (Y = 0). For example,
(14.4) CO, + H, wH,O + CO
Page 3
14
Chemical Kinetics
14.1 Introduction
Up until now it has been assumed that chemical reactions take place very rapidly, and that
the equilibrium conditions are reached instantaneously. While most combustion processes
are extremely fast, often their speed is not such that combustion can be considered to be
instantaneous, compared with the physical processes surrounding them. For example, the
combustion in a reciprocating engine running at 6000 rev/min has to be completed in
about 5 ms if the engine is to be efficient. Likewise, combustion in the combustion
chamber of a gas turbine has to be rapid enough to be completed before the gas leaves the
chamber. Such short times mean that it is not possible for all the gases in the combustion
chamber to achieve equilibrium - they will be governed by the chemical kinetics of the
reactions.
Chemical kinetics plays a major role in the formation of pollutants from combustion
processes. For example, oxygen and nitrogen will coexist in a stable state at atmospheric
conditions, and the level of oxides of nitrogen (NO,) will be negligible. However, if the
oxygen and nitrogen are involved in a combustion process then they will join together at
the high temperature to form NO, which might well be frozen into the products as the
temperature drops. This NO, is a pollutant which is limited by legislation because of its
initant effects. NO, is formed in all combustion processes, including boilers, gas turbines,
diesel and petrol engines; it can be removed in some cases by the use of catalytic
converters.
It was shown in Chapters 12 and 13 that significant dissociation of the normal products
of combustion, carbon dioxide and water can occur at high temperature. The values
shown were based on the equilibrium amounts of the substances, and would only be
achieved after infinite time; however, the rates at which chemical reactions occur are
usually fast and hence some reactions get close to equilibrium even in the short time the
gases are in the combustion zone. An analysis of the kinetics of reactions will now be
presented.
14.2 Reaction rates
Reaction rates are governed by the movement and breakdown of the atoms or molecules in
the gas mixture: reactions will occur if the participating ‘particles’ collide. The number of
Reaction rates 277
collisions occurring in a mixture will be closely related to the number densities (number
per unit volume) of the 'particles'. The number density can be defined by the molar
concentration, c, which is the amount of substance per unit volume. This is obviously a
measure of the number of particles per unit volume since each amount of substance is
proportional to the number of molecules; this is illustrated in Fig 14.1. The molar
concentration will be denoted by enclosing the reactant or product symbol in [ 1.
.. .
0.
.
. 0.
.O .
.
..
1: 'd.1
Fig. 14.1 Diagrammatic representation of molar concentration. Molar concentration in (a) is
approximately half that in (b)
Reactions occur when two, or more, reactants are capable of reacting. Many simple
chemical reactions are second order, e.g.
kf
A+B-C+D
kb
(14.1)
kf
C02 + H2-H20 + CO
kb
where the first reaction is a general one and the second is an example based on the water
gas reaction (chosen because the same number of reactants and products exist on both
sides).
These reactions can be written as
(14.2)
where A,, A,, .. . etc are the elements or compounds involved in the reaction, and
Y,, v2, . . . etc are the respective stoichiometric coefficients. Equation (14.2) can be further
generalised to
(14.3)
where q = total number of species being considered, v' represents the stoichiometric
coefficients of the reactants, and Y" that of the products.
In this case q has been taken as the same on both sides of the equation because it is
assumed that the same species can exist on both sides. If a species does not exist on one of
the sides it is represented by a stoichiometric coefficient of zero (Y = 0). For example,
(14.4) CO, + H, wH,O + CO
278 Chemical kinetics
can be represented by
4 b4
v:,~i-.C V;A,
i= 1 kb i=l
(14.5)
where q=4
vij = stoichiometric coefficient of i in reaction j
A,=C02 v\=l v’~=O
A, = H2 v;=1 v;=o
A,= CO vk=O v:=1
A,=H,O v;=O ~l;=l
The law of mass action, which is derived from the kinetic theory of gases, states that the
rate of formation, or depletion, of a species is proportional to the product of molar
concentrations of the reactants, each raised to the power of its stoichiometric coefficient.
Hence the rate of formation of species j, for an elementary reaction, is
Rj a [Al]”J[A2]’*J [A,]”
or
n
Rj=n[Ai]” for(j= 1, ... s)
i= 1
and s = total number of simultaneous reactions.
Equation (14.6) may be written as
n
Rj=k,n[AJyJ for(j= 1, ... s)
i= 1
(14.6)
(14.7)
where kj is the rate constantfor the reaction. In eqns (14.1) to (14.5) the reaction was
described as having a forward and backward direction, with rates k, and kb. The reaction
shown in eqn (14.7) can be described in this way as
n
R, = bj
[A,]’”
for (j = 1, . . . s) (14.8)
i= 1
and
n
Rbj = kbj n [A,]” for (j = 1,. . . S) (14.9)
The rate of change with time of species i is proportional to the change of the
stoichiometric coefficients of Ai in the reaction equation (the system is effectively a first
order one attempting to achieve the equilibrium state). Thus, for the forward direction
i= 1
(14.10)
Page 4
14
Chemical Kinetics
14.1 Introduction
Up until now it has been assumed that chemical reactions take place very rapidly, and that
the equilibrium conditions are reached instantaneously. While most combustion processes
are extremely fast, often their speed is not such that combustion can be considered to be
instantaneous, compared with the physical processes surrounding them. For example, the
combustion in a reciprocating engine running at 6000 rev/min has to be completed in
about 5 ms if the engine is to be efficient. Likewise, combustion in the combustion
chamber of a gas turbine has to be rapid enough to be completed before the gas leaves the
chamber. Such short times mean that it is not possible for all the gases in the combustion
chamber to achieve equilibrium - they will be governed by the chemical kinetics of the
reactions.
Chemical kinetics plays a major role in the formation of pollutants from combustion
processes. For example, oxygen and nitrogen will coexist in a stable state at atmospheric
conditions, and the level of oxides of nitrogen (NO,) will be negligible. However, if the
oxygen and nitrogen are involved in a combustion process then they will join together at
the high temperature to form NO, which might well be frozen into the products as the
temperature drops. This NO, is a pollutant which is limited by legislation because of its
initant effects. NO, is formed in all combustion processes, including boilers, gas turbines,
diesel and petrol engines; it can be removed in some cases by the use of catalytic
converters.
It was shown in Chapters 12 and 13 that significant dissociation of the normal products
of combustion, carbon dioxide and water can occur at high temperature. The values
shown were based on the equilibrium amounts of the substances, and would only be
achieved after infinite time; however, the rates at which chemical reactions occur are
usually fast and hence some reactions get close to equilibrium even in the short time the
gases are in the combustion zone. An analysis of the kinetics of reactions will now be
presented.
14.2 Reaction rates
Reaction rates are governed by the movement and breakdown of the atoms or molecules in
the gas mixture: reactions will occur if the participating ‘particles’ collide. The number of
Reaction rates 277
collisions occurring in a mixture will be closely related to the number densities (number
per unit volume) of the 'particles'. The number density can be defined by the molar
concentration, c, which is the amount of substance per unit volume. This is obviously a
measure of the number of particles per unit volume since each amount of substance is
proportional to the number of molecules; this is illustrated in Fig 14.1. The molar
concentration will be denoted by enclosing the reactant or product symbol in [ 1.
.. .
0.
.
. 0.
.O .
.
..
1: 'd.1
Fig. 14.1 Diagrammatic representation of molar concentration. Molar concentration in (a) is
approximately half that in (b)
Reactions occur when two, or more, reactants are capable of reacting. Many simple
chemical reactions are second order, e.g.
kf
A+B-C+D
kb
(14.1)
kf
C02 + H2-H20 + CO
kb
where the first reaction is a general one and the second is an example based on the water
gas reaction (chosen because the same number of reactants and products exist on both
sides).
These reactions can be written as
(14.2)
where A,, A,, .. . etc are the elements or compounds involved in the reaction, and
Y,, v2, . . . etc are the respective stoichiometric coefficients. Equation (14.2) can be further
generalised to
(14.3)
where q = total number of species being considered, v' represents the stoichiometric
coefficients of the reactants, and Y" that of the products.
In this case q has been taken as the same on both sides of the equation because it is
assumed that the same species can exist on both sides. If a species does not exist on one of
the sides it is represented by a stoichiometric coefficient of zero (Y = 0). For example,
(14.4) CO, + H, wH,O + CO
278 Chemical kinetics
can be represented by
4 b4
v:,~i-.C V;A,
i= 1 kb i=l
(14.5)
where q=4
vij = stoichiometric coefficient of i in reaction j
A,=C02 v\=l v’~=O
A, = H2 v;=1 v;=o
A,= CO vk=O v:=1
A,=H,O v;=O ~l;=l
The law of mass action, which is derived from the kinetic theory of gases, states that the
rate of formation, or depletion, of a species is proportional to the product of molar
concentrations of the reactants, each raised to the power of its stoichiometric coefficient.
Hence the rate of formation of species j, for an elementary reaction, is
Rj a [Al]”J[A2]’*J [A,]”
or
n
Rj=n[Ai]” for(j= 1, ... s)
i= 1
and s = total number of simultaneous reactions.
Equation (14.6) may be written as
n
Rj=k,n[AJyJ for(j= 1, ... s)
i= 1
(14.6)
(14.7)
where kj is the rate constantfor the reaction. In eqns (14.1) to (14.5) the reaction was
described as having a forward and backward direction, with rates k, and kb. The reaction
shown in eqn (14.7) can be described in this way as
n
R, = bj
[A,]’”
for (j = 1, . . . s) (14.8)
i= 1
and
n
Rbj = kbj n [A,]” for (j = 1,. . . S) (14.9)
The rate of change with time of species i is proportional to the change of the
stoichiometric coefficients of Ai in the reaction equation (the system is effectively a first
order one attempting to achieve the equilibrium state). Thus, for the forward direction
i= 1
(14.10)
Chemical kinetics of NO 279
and, for the backward direction
Hence the net rate of formation of Ai is
[A,]'' - kbj n [Ai]"
d [Ail j
i= 1
Consider the following rate controlled reaction equation:
kf
vaA + vbB-vcC + vdD
kb
The net rate of generation of species C is given by
-- d[C1 - [kf[A]'"[B]" - kb[c]''[DP]
dt
(14.11)
(14.12)
(14.13)
( 14.14)
Using the notation [A]/[Al, = a, [Bl/[B], = /3, [Cl/[Cl, = y and [Dl/[Dl, = 6, where
the suffix e represents equilibrium concentrations, gives
(14.15) d [ C ] /d t = k, a "0/3"b[A ] 2 [ B 12 - kbyVc 6 vd [ C 1,'. [ D ] 2
At equilibrium
k,[A]:[B]? = k,[C],'.[D],'d= Rj
Therefore the net rate is
(14.16)
d[C]/dt = Rj[av~/3vb - y"C6vd1 (14.17)
14.3 Rate constant for reaction, k
The rate constant for the reaction, k, is related to the ability of atoms or ions to combine.
In combustion engineering this will usually occur when two or more particles collide.
Obviously the collision rate is a function of the number of particles per unit volume, and
their velocity of movement, i.e. their concentration and temperature. Most chemical
reactions take place between two or three constituents because the probability of more than
three particles colliding simultaneously is too small. It has been found experimentally that
most reactions obey a law like that shown in Fig 14.2.
This means that the rate constant for the reaction can be defined by an equation of the
form
(14.18)
This equation is called the Arrhenius equation. The factor A is called the pre-
exponential factor, or the frequency factor, and is dependent on the rate at which
collisions with the required molecular orientation occur. A sometimes contains a
temperature term, indicating that the number of collisions is related to the temperature in
those cases. The term E in the exponent is referred to as the activation energy. The values
of A and E are dependent on the reactions being considered, and some values are
k = Ae - E/RT
Page 5
14
Chemical Kinetics
14.1 Introduction
Up until now it has been assumed that chemical reactions take place very rapidly, and that
the equilibrium conditions are reached instantaneously. While most combustion processes
are extremely fast, often their speed is not such that combustion can be considered to be
instantaneous, compared with the physical processes surrounding them. For example, the
combustion in a reciprocating engine running at 6000 rev/min has to be completed in
about 5 ms if the engine is to be efficient. Likewise, combustion in the combustion
chamber of a gas turbine has to be rapid enough to be completed before the gas leaves the
chamber. Such short times mean that it is not possible for all the gases in the combustion
chamber to achieve equilibrium - they will be governed by the chemical kinetics of the
reactions.
Chemical kinetics plays a major role in the formation of pollutants from combustion
processes. For example, oxygen and nitrogen will coexist in a stable state at atmospheric
conditions, and the level of oxides of nitrogen (NO,) will be negligible. However, if the
oxygen and nitrogen are involved in a combustion process then they will join together at
the high temperature to form NO, which might well be frozen into the products as the
temperature drops. This NO, is a pollutant which is limited by legislation because of its
initant effects. NO, is formed in all combustion processes, including boilers, gas turbines,
diesel and petrol engines; it can be removed in some cases by the use of catalytic
converters.
It was shown in Chapters 12 and 13 that significant dissociation of the normal products
of combustion, carbon dioxide and water can occur at high temperature. The values
shown were based on the equilibrium amounts of the substances, and would only be
achieved after infinite time; however, the rates at which chemical reactions occur are
usually fast and hence some reactions get close to equilibrium even in the short time the
gases are in the combustion zone. An analysis of the kinetics of reactions will now be
presented.
14.2 Reaction rates
Reaction rates are governed by the movement and breakdown of the atoms or molecules in
the gas mixture: reactions will occur if the participating ‘particles’ collide. The number of
Reaction rates 277
collisions occurring in a mixture will be closely related to the number densities (number
per unit volume) of the 'particles'. The number density can be defined by the molar
concentration, c, which is the amount of substance per unit volume. This is obviously a
measure of the number of particles per unit volume since each amount of substance is
proportional to the number of molecules; this is illustrated in Fig 14.1. The molar
concentration will be denoted by enclosing the reactant or product symbol in [ 1.
.. .
0.
.
. 0.
.O .
.
..
1: 'd.1
Fig. 14.1 Diagrammatic representation of molar concentration. Molar concentration in (a) is
approximately half that in (b)
Reactions occur when two, or more, reactants are capable of reacting. Many simple
chemical reactions are second order, e.g.
kf
A+B-C+D
kb
(14.1)
kf
C02 + H2-H20 + CO
kb
where the first reaction is a general one and the second is an example based on the water
gas reaction (chosen because the same number of reactants and products exist on both
sides).
These reactions can be written as
(14.2)
where A,, A,, .. . etc are the elements or compounds involved in the reaction, and
Y,, v2, . . . etc are the respective stoichiometric coefficients. Equation (14.2) can be further
generalised to
(14.3)
where q = total number of species being considered, v' represents the stoichiometric
coefficients of the reactants, and Y" that of the products.
In this case q has been taken as the same on both sides of the equation because it is
assumed that the same species can exist on both sides. If a species does not exist on one of
the sides it is represented by a stoichiometric coefficient of zero (Y = 0). For example,
(14.4) CO, + H, wH,O + CO
278 Chemical kinetics
can be represented by
4 b4
v:,~i-.C V;A,
i= 1 kb i=l
(14.5)
where q=4
vij = stoichiometric coefficient of i in reaction j
A,=C02 v\=l v’~=O
A, = H2 v;=1 v;=o
A,= CO vk=O v:=1
A,=H,O v;=O ~l;=l
The law of mass action, which is derived from the kinetic theory of gases, states that the
rate of formation, or depletion, of a species is proportional to the product of molar
concentrations of the reactants, each raised to the power of its stoichiometric coefficient.
Hence the rate of formation of species j, for an elementary reaction, is
Rj a [Al]”J[A2]’*J [A,]”
or
n
Rj=n[Ai]” for(j= 1, ... s)
i= 1
and s = total number of simultaneous reactions.
Equation (14.6) may be written as
n
Rj=k,n[AJyJ for(j= 1, ... s)
i= 1
(14.6)
(14.7)
where kj is the rate constantfor the reaction. In eqns (14.1) to (14.5) the reaction was
described as having a forward and backward direction, with rates k, and kb. The reaction
shown in eqn (14.7) can be described in this way as
n
R, = bj
[A,]’”
for (j = 1, . . . s) (14.8)
i= 1
and
n
Rbj = kbj n [A,]” for (j = 1,. . . S) (14.9)
The rate of change with time of species i is proportional to the change of the
stoichiometric coefficients of Ai in the reaction equation (the system is effectively a first
order one attempting to achieve the equilibrium state). Thus, for the forward direction
i= 1
(14.10)
Chemical kinetics of NO 279
and, for the backward direction
Hence the net rate of formation of Ai is
[A,]'' - kbj n [Ai]"
d [Ail j
i= 1
Consider the following rate controlled reaction equation:
kf
vaA + vbB-vcC + vdD
kb
The net rate of generation of species C is given by
-- d[C1 - [kf[A]'"[B]" - kb[c]''[DP]
dt
(14.11)
(14.12)
(14.13)
( 14.14)
Using the notation [A]/[Al, = a, [Bl/[B], = /3, [Cl/[Cl, = y and [Dl/[Dl, = 6, where
the suffix e represents equilibrium concentrations, gives
(14.15) d [ C ] /d t = k, a "0/3"b[A ] 2 [ B 12 - kbyVc 6 vd [ C 1,'. [ D ] 2
At equilibrium
k,[A]:[B]? = k,[C],'.[D],'d= Rj
Therefore the net rate is
(14.16)
d[C]/dt = Rj[av~/3vb - y"C6vd1 (14.17)
14.3 Rate constant for reaction, k
The rate constant for the reaction, k, is related to the ability of atoms or ions to combine.
In combustion engineering this will usually occur when two or more particles collide.
Obviously the collision rate is a function of the number of particles per unit volume, and
their velocity of movement, i.e. their concentration and temperature. Most chemical
reactions take place between two or three constituents because the probability of more than
three particles colliding simultaneously is too small. It has been found experimentally that
most reactions obey a law like that shown in Fig 14.2.
This means that the rate constant for the reaction can be defined by an equation of the
form
(14.18)
This equation is called the Arrhenius equation. The factor A is called the pre-
exponential factor, or the frequency factor, and is dependent on the rate at which
collisions with the required molecular orientation occur. A sometimes contains a
temperature term, indicating that the number of collisions is related to the temperature in
those cases. The term E in the exponent is referred to as the activation energy. The values
of A and E are dependent on the reactions being considered, and some values are
k = Ae - E/RT
280 Chemical kinetics
11 T
Fig. 14.2 Relationship between reaction rate and temperature
introduced below. The significance of the activation energy, E, is shown in Fig 14.3,
where it can be seen to be the energy required to ionise a particular molecule and make it
receptive to reacting. It appears in the exponential term because not all activated molecules
will find conditions favourable for reaction.
Activation
Enthalpy (or internal energy)
of reaction, Q, (or Q, )
Progress of reaction
Fig. 14.3 Schematic interpretation of activation energy
14.4 Chemical kinetics of NO
The chemical kinetics for the formation of NO are relatively well understood, and will be
developed here. The chemical kinetics for other pollutants can be derived in a similar way
if the necessary reaction rates are available, although it should be recognised that most
other pollutants are produced by reactions between the oxygen in the air and a constituent
of the fuel (e.g. carbon or sulfur). The formation of NO in combustion processes can occur
from two sources: thermal NO and prompt NO. Thermal NO is formed by the combination
of the oxygen and nitrogen in the air, and will be produced even if there is no nitrogen in
the fuel itself. This section will restrict itself to considering thermal NO. Prompt NO is
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