Combustion Chemistry
Hai Wang
Stanford University
2015 Princeton-CEFRC Summer School On Combustion
Course Length: 3 hrs
June 22 – 26, 2015
Copyright ©2015 by Hai Wang
This material is not to be sold, reproduced or distributed without prior written
permission of the owner, Hai Wang.
Stanford University
Version 1.2
©Hai Wang
Lecture 7
7. Laminar Premixed Flames
In this lecture we will provide a background of theory and simulation method for laminar
premixed flames. The basic theory of the steady, planar, one-dimensional gaseous reacting
flow will also be introduced.
7.1 Deflagration versus Detonation
Consider a steady, planar, one dimensional reacting front propagating from burned mixture
into an unburned mixture, as shown in Figure 7.1. For the time being, We shall not consider
chemical details. If we let the wave front velocity to be equal to zero (i.e., the location of the
planar flame is fixed), the unburned gas flows into the wave front at a velocity of v1. Because
of gas expansion accompanied by the temperature increase, the burned gas must leave the
wave front at a velocity v2 > v1. This setup represents, for example, a Bunsen flame in which
the flame front is made flat.
Alternatively we can have the unburned gas to assume a zero velocity, and the wave front
propagates into the unburned gas at the velocity v1; the burned gas exits at a velocity equal to
v1–v2. This velocity setup is equivalent to a flame wave propagating through a tube
(assuming a flat velocity profile in the radial direction).
The conservation equations may be given by
Mass:
m = ρ1v 1 A = ρ2v 2 A
(7.1)
Momentum:
P1 + ρ1v 12 = P2 + ρ2v 22
(7.2)
c pT1 +
Energy:
Equation of State:
v2
v 12
+ Q = c pT2 + 2
2
2
(7.3)
P1 = ρ1RT1 and P2 = ρ2RT2
(7.4)
where m is the mass flow rate, A is the area, ρ is the mass density, cp is the mean specific
heat, and R is the specific gas constant (J/g-K), Q is the reaction heat release per mass of
unburned gas (proportional to the reaction enthalpy −ΔH r,298 ), and the subscripts “1” and
“2” denote the unburned and burned mixtures, respectively. Obviously, several important
assumptions are made here. We have assumed that the specific heat of the mixtures may be
approximated as a constant, and its values for the burned and unburned mixtures are equal.
–v 2
–v 1
0
Wave front fixed
Burned gas
v1 – v2
Unburned gas
v1
0
Unburned gas velocity = 0
Wave front
7-1
Figure 7.1 Planar reacting wave
and velocities used in its analysis.
Stanford University
Version 1.2
©Hai Wang
We now wish to obtain the Hugoniot relation that governs the detonation versus
deflagration problems. Combining equations (7.1) and (7.2) and rearranging, we obtain
⎡ ρ ⎤
⎡1
1⎤
P2 − P1 = v 12 ρ1 ⎢1− 1 ⎥ = v 12 ρ12 ⎢ − ⎥ ,
⎣ ρ2 ⎦
⎣ ρ1 ρ2 ⎦
(7.5)
or
v 12 =
P2 − P1
.
1⎤
2⎡ 1
ρ1 ⎢ − ⎥
⎣ ρ1 ρ2 ⎦
(7.6)
Similarly, we also have
v 22 =
P2 − P1
.
1⎤
2⎡ 1
ρ2 ⎢ − ⎥
⎣ ρ1 ρ2 ⎦
(7.7)
We now put equations (7.4) into (7.3) and obtain
Q=
(
)
cp ⎛ P
P ⎞ 1
2
− 1 ⎟ − v 12 − v 22 .
⎜
R ⎝ ρ2 ρ1 ⎠ 2
(7.8)
Since γ = Cp/Cv and R = Cp –Cv, we have
cp
R
=
γ
γ −1
(7.9)
and
Q=
(
)
γ ⎛ P2 P1 ⎞ 1 2 2
−
− v −v .
γ − 1 ⎜⎝ ρ2 ρ1 ⎟⎠ 2 1 2
(7.10)
Combining equations (7.6), (7.7) and (7.10), one obtains the Hugoniot equation
Q=
⎛ 1
γ ⎛ P2 P1 ⎞ 1
1⎞
− ⎟ − ( P2 − P1 ) ⎜ + ⎟ .
⎜
γ − 1 ⎝ ρ2 ρ1 ⎠ 2
⎝ ρ1 ρ2 ⎠
For a simple shock wave, Q ≡ 0, and the Hugoniot equation becomes
7-2
(7.11)
Stanford University
Version 1.2
©Hai Wang
⎛ 1
γ ⎛ P2 P1 ⎞ 1
1⎞
− ⎟ − ( P2 − P1 ) ⎜ + ⎟ = 0 .
⎜
γ − 1 ⎝ ρ2 ρ1 ⎠ 2
⎝ ρ1 ρ2 ⎠
(7.12)
The above equation states that for a given unburned gas condition (P1, 1/ρ1), there exist a
family of solutions for (P2, 1/ρ2). The trivial solution is P1 = P2 and ρ1 = ρ2 and the
remaining possible values of (P2, 1/ρ2) for a given (P1, 1/ρ1) may be schematically depicted
as a hyperbola, as shown in Figure 7.2. Then, for an arbitrary Q > 0, the possible values of
(P2, 1/ρ2) may be represented by a finite displacement of the (P2, 1/ρ2) curve for Q = 0,
again shown in Figure 7.2. Of course, a family of Hugoniot curves may be obtained for
different values of Q.
One may identify five sections on a Hugoniot curve, some of which are physically realistic
but others are not. Points J and Y are obtained by drawing the tangents to either side of the
curve from the (P1, 1/ρ1) point. In addition, one may intersect the Hugoniot curve with
constant P1 and constant 1/ρ1 lines. The curve is then split in to regions I through V.
Consider region V, we have P2 > P1 and 1/ρ1 < 1/ρ2. It follows that
⎡1
1⎤
v 12 = P2 − P1 ρ12 ⎢ − ⎥ < 0 ,
⎣ ρ1 ρ2 ⎦
and region V is physically unrealistic.
Figure 7.2 Hugoniot curves with and without heat
release.
For regions I and II we have P2 > P1, and ρ2 > ρ1. In other words, the pressure and density
of the burned gas can be far greater than those of the unburned gas. The pressure difference
across the flame front therefore causes the production of a shock wave that propagates into
the unburned gas. Therefore, regions I and II correspond to conditions that lead to
detonation. It may be shown that region II is unstable with respect to wave propagation and
stable detonation waves are produced in region I.*
*
See, Glassman, I. Combustion, 2nd Ed., Academic Press, Orlando, FL, 1996, Chapter 5.
7-3
Stanford University
Version 1.2
©Hai Wang
Regions III and IV represent the deflagration phenomenon, in which the pressure of the
burned gas is just slightly smaller than the unburned gas, but the mass density of the burned
gas is far smaller than that of unburned gas (due to gas expansion). Again, region IV is
shown to be unstable, as discussed in detail by Glassman.*
Laminar premixed flames (e.g., Bunsen flames) are basically deflagration waves propagating
from the burned to unburned mixtures.
7.2 Qualitative Feature of Laminar Premixed Flames
Consider a planar, stationary, adiabatic premixed flame. A magnified structure across the
flame front may be viewed in Figure 7.3. The concentrations of the reactants in the
unburned gas are constant until the flame front, where they begin to be consumed and
products begin to form. Of course, we learned earlier that the reactants usually do not get
converted to products in one single chemical step. Rather a variety of combustion
intermediates are produced. For this reason, the concentrations of the intermediates from,
peak and then decay as the concentration of products reach their equilibrium values.
Likewise, the temperature starts to rise as the reactants enter into the flame front; and it
reaches the adiabatic flame temperature when reaction comes to equilibrium. Therefore a
basic understanding of an adiabatic laminar premixed flame provides us with further
chemical details into a chemical equilibrium process, as it also gives the rate at which the
final equilibrium is reached.
–v 2
0
–v 1
Burned gas
x
Wave front fixed
Unburned gas
reactants
products or T
intermediates
Figure 7.3 Schematic structure of a
planar, stationary, adiabatic premixed
flame.
x
0
Reaction zone
An examination of the structure inside the flame front tells us that to understand a laminar
premixed flame, we need to address the problems of conductive heat transfer, from the
burned gas to the unburned gas, mass transfer due to Fickian diffusion of species, and
intense chemical reaction inside and ahead of the flame front. Since a fuel-air mixture
contains at least 3 species in the reactant stream and 3 products in the products, we need to
consider multi-component transport of heat and chemical species.
7-4
Stanford University
Version 1.2
©Hai Wang
7.3 Governing Equations
The conservation equations describing a steady, planar, adiabatic premixed flame are given
by
m = ρvA = constant
Continuity:
Species:
Energy: ρvAc p
ρvA
(7.13)
dYk d
+ ( ρ AYkVk ) − Aω kWk = 0
dx dx
(7.14)
dT d ⎛
dT ⎞
dT
K
K
− ⎜ λ A ⎟ + A ∑ k=1 ρYkVkc p,k
− A ∑ k=1ω khkWk = 0 , (7.15)
dx dx ⎝
dx ⎠
dx
where x denotes the spatial coordinate, Vk is the diffusion velocity of species k, λ is the
mixture conductivity, cp is the mixture specific heat, and cp,k is the specific heat of the kth
species. The molar production rate ω has already been given by equation (6.6).
k
The boundary conditions of a steady, planar, adiabatic, freely propagating, premixed flame are
that the temperature at the cold boundary is equal to the unburned gas temperature,
T = Tu
for x → −∞ .
(7.16)
and the temperature gradient vanishes at the hot boundary
dT
=0
dx
for
x→∞.
(7.17)
The species boundary conditions are specified similarly, by
Yk = Yk,0 (k = 1,...K )
for
x → −∞ ,
(7.18)
and
dYk
= 0 (k = 1,...K )
dx
for x → ∞ .
(7.19)
These governing equations along with their boundary conditions form a two-point boundary
value problem, which may be solved numerically if the molar production rates ω are
k
specified.
In examining these equations, however, one realizes that the mass flow rate m (or linear
velocity v) is not known, and must be solved as an eigenvalue. Physically, equations (7.13)
through (7.15) state that the flame front is not anchored; and an infinite number of solutions
may be obtained by adding a constant to the special coordinate x. This condition affords us
7-5
Stanford University
Version 1.2
©Hai Wang
to specify one more boundary condition to anchor the location of the flame. This is usually
done by explicitly specifying a particular temperature at a given spatial position xa, i.e.,
T ( x a ) = T1 > Tu .
(7.20)
Because we solve the aforementioned problem in a doubly infinite spatial domain, the
solution is adiabatic as no conductive heat loss can occur in either side of the boundary
where dT/dx = 0. It follows that without explicitly carrying out equilibrium calculations, the
temperature at the hot boundary T ( x → ∞ ) must be equal to the adiabatic flame
temperature Tf. In addition, the species concentrations Yk ( x → ∞ ) must also represent the
equilibrium composition.
There is a second type of laminar premixed flames, known as the burner stabilized flame.
Figure 7.4 shows a burner-stabilized premixed flame where the one-dimensional region of
the flame is masked by the yellow luminosity resulting from light emission of soot. In these
flames the stabilization is accomplished with heat loss to the burner. For these flames, the
mass flow rate m is known, and the temperature and mass flux fraction are specified at the
cold boundary and vanishing gradients are imposed at the hot boundary.
Figure 7.4 A sample burner-stabilized laminar premixed flame.
The shape of the flame is masked here by light emission due to
soot emission. The one-dimensional region is shown within
the rectangular box. Above the rectangle, the one-dimensional
flame is distorted by buoyancy.
7.4 Mass Transfer through Diffusion and Thermal Diffusion
In equations (7.14) and (7.15) the diffusion velocity Vk is composed of two parts, the
ordinary (Fickian) diffusion velocity VD,k and thermal diffusion velocity VT,k,
Vk = VD,k +VT ,k .
(7.16)
Fickian diffusion is the result of spatial gradient in species concentration. Our experience
tells us that for a gas of spatially non-uniform species concentration, the random walk of gas
molecules would eventually even out the spatial non-uniformity. The rate of this process is
measured by the diffusion flux j (mol/m2-s) of a species with respect to the mass center of a
gas,
7-6
Stanford University
Version 1.2
©Hai Wang
j k = −Dk
∂c k
,
∂x
(7.17)
where Dk (cm2/s) is the diffusivity of species k and ck is the molar concentration. Dividing
equation (7.17) by the total molar concentration c (i.e., under constant temperature and
pressure) and recognizing that Xk = ck/c, one obtains
VD,k = −Dk
1 dX k
,
X k dx
(7.18)
for a one-dimensional flow.
For a binary mixture, we have
j 1 = −D1
∂c 1
,
∂x
(7.19)
j 2 = −D2
∂c 2
.
∂x
(7.20)
Since mass conservation requires that under the steady condition j1 = –j2 and ∂c ∂x = 1
∂c ∂x , we have D1 = D2. These diffusion coefficients are known as the binary diffusion
2
coefficients, denoted by D12 = D21. To the first approximation, the mixture-average
diffusivity may be given by
Dk =
1− Yk
∑ j ≠k X j Dkj
K
,
(7.21)
The thermal diffusion velocity arises from non-uniform temperature. Consider two types of
molecules with molecular mass m2 > m1. Under a local temperature gradient, a heavier
molecule would undergo stronger collision and greater momentum transfer on the “hot side”
of the molecule than on its “cold side”. It follows that the heavier molecule would
preferentially move to the low-temperature region,* whereas by mass conservation we easily
conclude that the light molecules would have to move to the high temperature region—a
phenomenon termed thermal diffusion (or the Soret effect). The velocity generated from
light molecules moving to hot spatial region is given by the thermal diffusion velocity.
To a first approximation, the thermal diffusion velocity may be given by
*
This phenomenon is sometime called thermophoresis.
7-7
Stanford University
Version 1.2
©Hai Wang
VT ,k = −
where k
T ,k
DkkT ,k 1 dT
,
X k T dx
(7.22)
is the thermal diffusion ratio.
7.5 Simplified Analysis of Laminar Premixed Flames
A qualitative understanding of a laminar premixed flame may be achieved by simplifying the
governing equations (7.13-15) and solving for the dependence of the mass flow rate with
respect to reaction rate. Here we assume that (i) the flame area A is a constant; (ii) all gas
properties, including the transport and thermal properties, are constant; thermal diffusion
may be neglected; (iii) heat transfer is accomplished entirely through heat conduction; and (iv)
the chemical reaction may be represented by a single rate step of reactant products with a
fixed heat release Q per mass of reactant consumed. We write
Continuity:
m ′ = m A = ρv = constant
(7.13)
dY
d 2Y
− ρ D 2 − ω = 0
dx
dx
(7.23)
dT
d 2T
− λ 2 − ω Q = 0 .
dx
dx
(7.24)
Species:
m ′
Energy:
m ′c p
Here m ′ is the mass flux (kg/m2-s), Y is the mass fraction of the reactant, and ω (kg/m3-s)
is the mass rate of reactant destruction.
The boundary conditions for these equations are
Y = Y0 , T = Tu
for x → −∞ ,
(7.25)
Y = 0, T = T f
for x → ∞ .
(7.26)
and
We start from the energy equation and note that equations (7.23) and (7.24) are basically
identical to each other if ρD is equivalent to λ/cp. We shall divide the flame front into two
zones. In the first zone (I), we assume that the heat release is negligible (see Figure 7.5) and
the temperature variation is entirely governed by heat conduction. Accordingly the energy
equation is simplified to
dT
λ d 2T
−
= 0.
dx m ′c p dx 2
7-8
(7.27)
Stanford University
Version 1.2
©Hai Wang
with boundary conditions given by
T = Tu
for x → −∞ ,
(7.28)
for x = 0 ,
(7.29)
T = Ti
where Ti is the temperature at the end of the heat conduction zone. Equation (7.27) has the
solution of the form
dT m ′c p (T − Tu )
=
.
dx
λ
(7.30a)
m ′c p (Ti − Tu )
⎛ dT ⎞
.
⎜⎝ dx ⎟⎠ =
λ
0
(7.30b)
or
T
Zone I
(conduction
Region)
Tf
Ti
δ
Zone II (heat release region)
Tu
0
Figure 7.5 A simplified description of
temperature variation in a laminar
premixed flame.
x
In zone II with its width given by δ, we assume that the heat release rate is not negligible,
but the convective heat transfer is unimportant,
d 2T
dx
2
+
ω Q
=0.
λ
(7.31)
The boundary conditions of the above differential equation are
T = Ti
for x = 0 ,
(7.32)
T = Tf
for x = δ .
(7.33)
The solution of equation (7.31) is obtained in the following manner. We have
7-9
Stanford University
Version 1.2
©Hai Wang
2
dT d 2T 1 d ⎛ dT ⎞
ω Q dT
=
=−
.
⎜
⎟
2
dx dx
2 dx ⎝ dx ⎠
λ dx
(7.34)
It follows that
⎛ Q
⎛ dT ⎞
=
⎜2
⎜⎝ dx ⎟⎠
⎜⎝ λ
x =0
∫
Tf
Ti
⎞
ω dT ⎟
⎟⎠
12
.
(7.35)
Equating (7.30b) and (7.35), we obtain
⎛ Q
λ
m ′ =
⎜2
c p (Ti − Tu ) ⎜⎝ λ
∫
Tf
Ti
⎞
ω dT ⎟
⎟⎠
12
.
(7.36)
7.6 The Laminar Flame Speed
Let us define the laminar flame speed s uo to be the linear velocity of the unburned gas and
m ′ = ρ0 s u . Substituting the definition for s uo into (7.36) and define the thermal diffusivity to
be α = λ ρ0c p , we have
⎛ Q
α
s u =
⎜2
(Ti − Tu ) ⎜⎝ λ
∫
Tf
Ti
⎞
ω dT ⎟
⎟⎠
12
.
(7.37)
The total heat release Q may be equated with the enthalpy used to heat up the burned gas,
i.e.,
(
)
Q = c p T f − Tu .
(7.38)
Assuming that Ti is very close to Tf, we re-write equation (7.37) as
⎡
⎢ 2α
s u ≈ ⎢
⎢ T f − Tu
⎣
(
Tf
)
∫T
i
ω dT ⎤⎥
⎥
ρ0 ⎥
⎦
12
.
(7.39)
Clearly, the laminar flame speed is proportional to the square root of the thermal diffusivity.
Empirically, the mass destruction rate ω is of the order of
7-10
Stanford University
Version 1.2
©Hai Wang
⎛ E ⎞
ω = A ' exp ⎜ − a ⎟ .
⎝ R uT ⎠
(7.40)
Since we have assumed that most of the heat release occurs within a small distance δ, and Tf
~ Ti, we can assume a variable θ, such that for T > Ti
θ = T − Ti  T f .
(7.41)
It follows that
Ea
Ea
=
.
R u T R T 1− θ T
u f
f
(
(7.42)
)
The above equation may be approximated as
(
)
Ea
Ea
Ea
θ Ea
.
=
1+ θ T f =
+
R uT R uT f
R u T f R u T f2
(7.43)
Let
β=
θ Ea
R u T f2
.
(7.44)
⎞
⎟ exp ( − β ) .
⎠
(7.45)
One obtains
⎛ E
ω = A ' exp ⎜ − a
⎝ R uT f
and
Tf
∫T
i
ω dT
ρ0
⎛ E ⎞ R u T f2 βi
A'
=
exp ⎜ − a ⎟
∫ exp ( − β ) d β
ρ0
⎝ R u T f ⎠ Ea 0
.
⎛ E ⎞ R u T f2
A
=
exp ⎜ − a ⎟
1− e − βi
ρ0
⎝ R u T f ⎠ Ea
(
'
(7.46)
)
(
Since in general Ea/RuT > 1 and θ T f ~ 0.1 , β is of the order of unity and 1− e
i
It follows that
7-11
− βi
) ~ 1.
Stanford University
Version 1.2
©Hai Wang
⎡
⎛ E ⎞
R u T f2
A'

a
⎢
s u ≈ 2α
exp ⎜ −
⎟
⎢ ρ0
R uT f ⎠ E T − T
⎝
a
f
u
⎣
(
)
⎤
⎥
⎥
⎦
12
.
(7.47)
The above description of the laminar flame speed is only qualitative, but it nonetheless tells
us several important factors that govern the flame speed of a combustible mixture.
Specifically the flame speed is proportional to the square root of thermal diffusivity.
Therefore one of the limiting factors for the flame speed is how fast the heat can be
transferred from the reaction zone, where heat is release, to the unburned mixture. Since the
thermal diffusivity is inversely proportional to mass density and thus pressure, the flame
speed has roughly a P–1/2 dependence.
What is perhaps more important is the flame temperature. Equation (7.47) shows that the
flame speed is largely governed by the flame temperature because of the exponential
function. Indeed the flame speed follows a dependence on the fuel equivalence ratio in a
fashion similar to that between the adiabatic flame temperature and the fuel equivalence ratio,
as seen in Figures 7.6 and 7.7. Namely, the flame speed of hydrocarbon fuels peaks around
the equivalence ratio of unity and the peak value occurs slightly on the fuel rich side.
Note that for alkane fuels (e.g., propane and n-butane) the maximum flame speed is about 40
cm/s at the ambient pressure, whereas for some more energetic fuels (e.g., acetylene and
ethylene), the maximum flame temperature at the ambient pressure is larger (~1 m/s).
Adiabatic Flame Temperature, T f (K)
Laminar Flame Speed, s u o (cm/s)
160
140
Acetylene (C2H2)
120
100
80
60
Ethylene (C 2H4)
40
20
n -Butane (C 4H10)
0
0.5
1.0
Propane (C3H8)
1.5
Acetylene (C2H2)
2500
Ethylene (C2H4)
2000
Propane (C3H8)
n -Butane (C 4H10)
1500
0.5
2.0
1.0
1.5
2.0
Equivalence Ratio, φ
Equivalence Ratio, φ
Figure 7.6 Variation of the laminar flame speed
of selected fuel-air mixtures at the atmospheric
pressure as a function of the equivalence ratio.
Symbols are actual experimental data taken from
literature and lines are computed using detailed
reaction mechanism.
Figure 7.7 Variation of the adiabatic flame
temperature selected fuel-air mixtures at the
atmospheric pressure as a function of the
equivalence ratio.
7-12
Stanford University
Version 1.2
©Hai Wang
The combined effect of thermal diffusivity and flame temperature may be viewed by
examining the flame speed of hydrogen-oxygen-diluent mixtures. One of the unique
features of hydrogen is that under the stoichiometric condition the molar ratio of hydrogen
to oxygen is 2 to 1 (compared with the stoichiometric molar ratio of hydrocarbon to oxygen
which is typically smaller than 0.5). Therefore the hydrogen-air mixture usually contains a
large amount of hydrogen especially under the fuel rich condition. Owing to a small specific
heat and more importantly a small mass density of hydrogen, the thermal diffusivities of
both burned and unburned hydrogen-air mixtures are usually much larger than those of
hydrocarbon-air mixtures. This not only causes hydrogen-air mixtures to assume peak flame
speed value (~2.5 m/s) substantially large than those of hydrocarbon-air mixtures, it also
leads to the peak flame speed to shift itself far into the fuel rich condition (see, Figure 7.8),
compared to the adiabatic flame temperature profiles shown in Figure 7.9.
The effect of thermal diffusivity on the flame speed may be further amplified when we
examine hydrogen-air like mixtures, in which the nitrogen is replaced by noble gases helium
(He) and argon (Ar), as seen in Figure 7.8. Because monoatomic gases like helium and argon
have small specific heats, the adiabatic flame temperature of the resulting hydrogen-oxygeninert mixtures is higher than that of the hydrogen-air mixture (see, Figure 7.9). Likewise the
flame speed for noble gas substituted mixtures is also larger than that for the basic air
mixtures. On the other hand, we also notice that the flame speed of the helium mixture is
larger than that of the argon mixture, despite the fact that the adiabatic flame temperature is
identical. Clearly the differences in the flame speed values are caused by the smaller mass
density associated with helium and consequently the larger thermal diffusivity of hydrogenoxygen-helium mixtures.
Adiabatic Flame Temperature, T f (K)
Laminar Flame Speed, s u o (cm/s)
3000
400
He
300
Ar
200
N2
100
0
2500
He & Ar
2000
1000
0
1
2
3
4
N2
1500
0
1
2
3
4
Equivalence Ratio, φ
Equivalence Ratio, φ
Figure 7.8 Variation of the laminar flame speed of
hydrogen-air and hydrogen-air like mixtures at the
atmospheric pressure as a function of the
equivalence ratio. Symbols are actual experimental
data taken from literature and lines are computed
using detailed reaction mechanism of Davis et al.
(see, Figure 3.7/Lecture 3 notes).
7-13
Figure 7.9 Variation of the adiabatic flame
temperature of hydrogen-air and hydrogen-air like
mixtures at the atmospheric pressure as a function
of the equivalence ratio.
Stanford University
Version 1.2
©Hai Wang
7.7 More Detailed Analysis of Laminar Premixed Flames
The discussion in the preceding section shows that to the first approximation the intensity of
laminar premixed flames is influenced by the thermal diffusivity and adiabatic flame
temperature. At the more detailed level it is known that the laminar flame speed is
influenced by the heat release rate and thus the detailed chemistry or reaction kinetics. In
addition, the diffusivities of the fuel and of reactive intermediates, such as the hydrogen
atom, can also influence the flame speed. The overall chemistry effect is exhibited in the A
factor and the activation energy of equation (7.47). Beyond a phenomenological description
of laminar flames, as discussed in section 7.6, and to quantitatively predict the flame speed,
we must consider detailed reaction mechanisms and rates, based on the knowledge presented
in the previous lecture. Therefore the quantitative solution of the laminar flame speed can
only come by solving equations (7.13-15). The use of detailed reaction models to predict the
flame speed and flame structures will be discussed in a later section.
A consideration of the detailed chemistry also means that we need accurate transport
coefficients. These include the mixture conductivity and species diffusivity, both of which
appear in the governing equations (7.14) and (7.15). Below, we shall give a brief review of
the molecular transport theory.
7.7.1 Theory of Molecular Transport
The standard theory of molecular transport is that of the Chapman-Enskog theory,
developed independently by Enskog in 1911 and Chapman in 1917.* Before we have a more
in-depth discussion of this theory, let us examine a simple gas-kinetic theory description of
viscosity, thermal conductivity and diffusivity of low-density gases.
Consider the motion of fluid between two plates (Figure 7.10). The lower plate is kept at
rest and the upper plate is moving at a velocity U in the x direction. The gas immediately
adjacent to the lower plate would assume a mean velocity equal to zero, i.e., u(0) = 0, and the
gas just below the upper plate has a velocity u(L) = U. Under the steady state, the viscous
force within the gas would cause the gas to establish a linear velocity gradient in the z
direction, i.e., u(z) = Uz/L, provided that L is much larger than the mean free path of the
gas. Note that u(z) is the unidirectional, mean velocity of the gas, and the gas molecules
themselves undergo random motion in all directions.
u(L) = U
z
L
0
Figure 7.10 A simple geometry
for measuring gas viscosity.
u(0) = 0
x
See, S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Third
Edition, Cambridge Press, Cambridge, 1995.
*
7-14
Stanford University
Version 1.2
©Hai Wang
The viscosity of a gas is, after all, momentum transport due to the random motion of
molecules and molecular collision in the z direction. Consider a molecule having velocity v
= (vx, vy, vz). Its spatial position is z′ when it undergoes the last collision. The distance it
traveled before the next collision occurs is, on average, the mean free path λg (see,
Homework assignment #4). Then we have (z′–z)/λg = –vz/v or z′ = z – λgvz/v. In other
words, the last collision occurs from below if vz > 0. Let us assume that the molecule has
come to local equilibrium after the last collision with its local environment, and it carries the
momentum at z′ equal to p(z′) =p(z – λgvz/v). We now expand p(z′) in the form of
λ g v z dp
+ .
v dz
p (z ′ ) = p (z ) −
(7.48)
The number of molecules with velocity v at position z′ is fv(vx,vy,vz)Ndvxdvydvz,* where fv(vx,vy,vz)
is the Boltzmann distribution of velocity (see, equation 4.14) and N is the total number
density. Since the molecules have a non-zero mean velocity in the x direction, the
Boltzmann distribution of energy is written as
(
f v x ,v y ,v z
)
⎛ m ⎞
=⎜
⎟
⎝ 2π kBT ⎠
32
(
)
2
⎧
⎡
2
2⎤⎫
⎪⎪ m ⎢⎣ v x − u (z ) + v y + v z ⎥⎦ ⎪⎪
exp ⎨ −
⎬.
2kBT
⎪
⎪
⎪⎩
⎪⎭
(7.49)
The flux of molecules crossing z′ is vzfv(vx,vy,vz)Ndvxdvydvz, each of which carries a momentum
equal to p(z′). The net momentum flux is obtained by integrating over all velocities,
∞
p =
∫ ∫ ∫ p (z ′)v f (v ,v ,v ) Ndv dv dv .
z v
x
y
z
x
y
(7.50)
z
−∞
Neglecting the higher order terms of equation (7.48) and substituting it into the above
equation gives a sum of two integrals,
p ≈ p (z ) N
−λ g N
*
∞
∫ ∫ ∫ v f (v ,v ,v )dv
dp
dz
z v
−∞
∞
x
y
z
x
dv y dv z
.
v z2
∫ ∫ ∫ v f (v ,v ,v )dv dv dv
v
x
y
z
x
y
z
−∞
Or rather, the molecules residing in x to x+dx, y to y+dy, and z′+dz′ is f(vx,vy,vz)Ndvxdvydvz.
7-15
(7.51)
Stanford University
Version 1.2
©Hai Wang
It can be shown that the first integral is zero.** For the second integral, we can assume that
u (z )  v x , because vx is of the order of the sonic velocity and u(z) is generally much smaller
than the sonic velocity. It follows that
dp
p ≈ − λ g N
dz
= −λ g N
∞
∫∫∫
−∞
v z2 ⎛ m ⎞
v ⎜⎝ 2π kBT ⎟⎠
32
⎧ m ⎡v 2 + v 2 + v 2 ⎤ ⎫
x
y
z ⎦⎪
⎪
exp ⎨ − ⎣
⎬ dv x dv y dv z
2kBT
⎪⎩
⎪⎭
,
(7.52)
dp ⎛ 1 ⎞
v
dz ⎜⎝ 3 ⎟⎠
where v is the mean velocity of the gas (see, equation 4.18). Since the momentum is
proportional to mean velocity, we have
du (z )
dp
=m
,
dz
dz
(7.53)
du (z )
1
p ≈ − λ g mNv
.
3
dz
(7.54)
and
Comparing the above equation with Newton’s law of viscous flow,
p = −η
du (z )
,
dz
(7.55)
where η is the viscosity, we obtain an approximate formula for the viscosity
1
η ≈ λ g mNv
3
.
2 mkBT 1
=
3
π πσ 2
(7.56)
The above equation predicts that the viscosity is independent of pressure.
**
That is,
∞
∫∫∫ (
−∞
)
⎛ m ⎞
v z f v x ,v y ,v z dv x dv y dv z = ⎜
⎝ 2π kT ⎟⎠
32
∫
∞ −
−∞
(
m v x −u (z )
e
)
2kBT
2
dv x
∫
∞ −
−∞
e
mv 2y
2kBT
dv y
∫
∞
−∞
vz e
−
mv z2
2kBT
dv z
where the last integral is
⎛ mv 2 ⎞
z
v z exp ⎜ −
⎟dv =
⎜⎝ 2kBT ⎟⎠ z
−∞
∫
∞
∫
∞
0
vz e
−
mv z2
2kBT
dv z +
∫
0
−∞
vz e
7-16
−
mv z2
2kBT
dv z =
∫
∞
0
vz e
−
mv z2
2kBT
dv z −
∫
∞
0
vz e
−
mv z2
2kBT
dv z = 0 .
Stanford University
Version 1.2
©Hai Wang
The procedure used in the above discussion is also applicable to thermal conductivity. All
what we need to do is to replace the momentum by sensible heat cv T and the momentum
flux by heat flux to obtain
dT (z )
c
1
q ≈ − λ g N v v
,
3
N avg
dz
(7.57)
where N avg is the Avogadro number. It follows that the thermal conductivity may be given
approximately by
c
1
λ ≈ λ g Nv v
3
N avg
=
2 cv
3 N avg
=
cv η
N avg m
kBT 1
.
π m πσ 2
(7.58)
The above equation states that the thermal conductivity is proportional to the heat capacity,
and the mean velocity of molecular random motion, and inversely proportional to the
collision cross section. The conductivity is again independent of pressure.
Equation (7.58) is only approximate, since for one, it does not consider the collision energy
transfer between a slightly hotter molecule traveling from a high-temperature region with a
slightly cooler molecule traveling from a cooler region. A more rigorous consideration
would lead us to conclude that on the one hand molecular collision can make heat
conduction slower because the cooler molecules effectively block the motion of hotter
molecules traveling into the cooler region, on the other hand molecular collision can increase
the thermal conductivity because of energy transfer during collision.
Lastly, we wish to consider the binary diffusion coefficient using the method similar to the
derivations of the approximate equations of the viscosity and thermal conductivity. An
idealized experiment to measure the binary diffusion coefficient would be a tube filled with
two gases of types 1 and 2. Let the tube be collinear with the z axis, where a concentration
gradient exists. The total number density is a constant, i.e.,
N = N 1 (z ) + N 2 (z ) = constant .
The number density of molecules arriving at z′ after the last collision is
7-17
(7.59)
Stanford University
Version 1.2
©Hai Wang
N 1 (z ′ ) = N 1 (z ) − λ g
v z dN 1 (z )
v
dz
+ ,
(7.60)
and the number of molecules passing through a unit plane perpendicular to the z axis is
approximately
v z N 1 (z ′ ) f 1,v ≈ v z N 1 (z ) f 1,v − λ g
dN 1 (z ) v z
2
dz
v
f 1,v .
(7.61)
In the above equations, the mean free path may be considered as the average of the mean
free paths of the two types of molecules. We obtain the net flux of type 1 molecules by
integrating the above equation over all velocities, which gives
j1 ≈ −λ g
dN 1 (z )
dz
∞
v z2
∫ ∫ ∫ v f (v ,v ,v )dv dv dv
v
x
y
z
−∞
dN (z )
1
= − λ g v1 1
3
dz
x
y
z
.
(7.62)
Comparing to the Fick’s law of diffusion, we obtain
1
D12 ≈ λ g v1 .
3
(7.63)
For molecules of types 1 and 2 having almost identical collision diameter, we have
D12 ≈
2 kBT 1 1
.
2
3 π m N πσ 12
(7.64)
Thus, the binary diffusion coefficient is inversely proportional to number density or pressure,
as expected.
Equations (7.56), (7.58), and (7.64) are useful, in that they give the correct temperature and,
if any, pressure dependencies for the viscosity, thermal conductivity, and diffusion
coefficient, respectively. A more rigorous treatment of these transport parameters will have
to consider the effects of collision energy transfer and the molecular force of interactions.
These were attempted by Chapman and Enskog.* The method involves an expansion of the
reduced distribution function in the phase space (v,r,t) then combined it with the Boltzmann
See, S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Third
Edition, Cambridge Press, Cambridge, 1995; J. O. Hirschfelder, C. F. Curtiss and R. B. Bird,
Molecular Theory of Gases and Liquids, John Wiley & Sons, New York, 1954.
*
7-18
Stanford University
Version 1.2
©Hai Wang
equation to obtain the hydrodynamic equations. The final results for a pure species labeled
as k is
5 π m kkBT
.
16 πσ 2Ω(2,2)*
(7.65)
λk =
π m kkBT 1 5 cv ηk
25 cv
=
.
32 N avg πσ 2Ω(2,2)* m 2 N avg m
k
(7.66)
D jk =
kBT
3 2π kBT
1+ Δ jk .
16
µ jk Pπσ 2 Ω(1,1)*
jk
ηk =
k
(
)
(7.67)
where µjk is the reduced mass of the species pair j and k, Δjk is a higher order term, which is
l ,s *
approximately unity for all but small species like H•, H2 and He, and Ω( ) is the reduced
collision integral. If molecular interactions are described by the Lennard-Jones (12-6)
potential function (see, equation 4.21) and we define the following reduced properties,
Reduced intermolecular distance:
r* = r σ ,
(7.69)
Reduced impact parameter:
b* = b σ .
(7.70)
Reduced potential energy:
ϕ * = VvDW ε .
(7.71)
Reduced temperature:
T * = kBT ε .
(7.72)
Reduced (relative) kinetic energy:
g *2 =
1 2
µg ε .
2
(7.73)
the collision integral may be expressed by
( )
l ,s *
Ω( ) T * =
4
⎡ 1+ −l ⎤
( ) ⎥ T (*s +2
(s + 1)! ⎢1−
⎢
2 (1+ l ) ⎥
⎣
⎦
l
∞
∞
0
0
∫ ∫
)
e−g
*2
T * *2 s +3
g
(1− cos χ )b db dg
l
*
*
*
,
(7.74)
where the angle of deflection χ is given by equation (4.29). Equations (7.69) and (7.74) are
applicable for collision of two molecules that are alike and dislike. For dislike molecules, σ
and ε are replaced by mean values σjk and εjk, as given by equations (4.22) and (4.23).
Equation (7.74) states that the reduced collision integral is completely defined if the reduced
temperature is known. In practice the reduced collision integral values are tabulated as a
7-19
Stanford University
Version 1.2
©Hai Wang
function of T*, as shown in Table 7.1. Table 7.2 lists the Lennard-Jones collision diameters
and well depth for selected combustion species. Most of the potential parameter values were
derived from gas viscosity data, and when such data are not available, they are estimated. A
very good source for the transport properties and their estimates may be found in R. C.
Reid, R. C., J. M., Plausnitz, B. E., Poling The properties of Gases and liquids, 4th ed, McGrawHill, New York, 1987.
In addition to the treatment given above, the standard procedure is to consider the influence
of the dipole-dipole interactions in the potential energy. The treatment is detailed in the
document attached to this lecture notes (R. J. Kee, J. Warnatz, M. E. Coltrin, and J. A. Miller,
A FORTRAN computer code package for the evaluation of gas-phase, multicomponent transport properties,
Sandia Report 86-8246) gives a more detailed description of the formula used for reacting
flow simulations. Some contemporary issues of binary diffusion coefficients are discussed in
the supplemental readings, also attached in this lecture notes.
7-20
Stanford University
Version 1.2
©Hai Wang
l ,s *
Table 7.1 The integrals Ω( ) as a function of reduced temperature*
Taken from J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids,
John Wiley & Sons, New York, 1954.
*
7-21
Stanford University
Version 1.2
©Hai Wang
l ,s *
Table 7.1 The integrals Ω( ) as a function of reduced temperature (continued)
7-22
Stanford University
Version 1.2
©Hai Wang
Table 7.2 Lennard-Jones (12-6) potential parameters for selected species.
e/kB (K)
10.2
136.5
145
38
572.4
107.4
107.4
97.53
80
107.4
80
σ (Å)
2.576
3.33
2.05
2.92
2.605
3.458
3.458
3.621
2.75
3.458
2.75
C
CH
CH2
CH2*
CH3
CH4
71.4
80
144
144
144
141.4
3.298
2.75
3.8
3.8
3.8
3.746
CO
CO2
HCO
CH2O
CH2OH
CH3O
CH3OH
98.1
244
498
498
417
417
481.8
3.65
3.763
3.59
3.59
3.69
3.69
3.626
C2
C 2H
C 2H 2
H2CC
C 2H 3
C 2H 4
C 2H 5
C 2H 6
97.53
209
209
209
209
280.8
252.3
252.3
3.621
4.1
4.1
4.1
4.1
3.971
4.302
4.302
C 2O
HCCO
HCCOH
CH2CO
CH2CHO
C2H2OH
CH3CHO
CH3CO
232.4
150
436
436
436
224.7
436
436
3.828
2.5
3.97
3.97
3.97
4.162
3.97
3.97
Species
He
Ar
H
H2
H 2O
H 2O 2
HO2
N2
O
O2
OH
Species
C 3H 2
C 3H 3
aC3H4
pC3H4
aC3H5
C 3H 6
C 3H 7
iC3H7
nC3H7
C 3H 8
e/kB (K)
209
252
252
252
266.8
266.8
266.8
266.8
266.8
266.8
σ (Å)
4.1
4.76
4.76
4.76
4.982
4.982
4.982
4.982
4.982
4.982
C 4H
C 4H 2
n-C4H3
C 4H 4
n-C4H5
C 4H 6
C 4H 7
iC4H7
C 4H 8
iC4H8
tC4H9
iC4H9
pC4H9
sC4H9
C4H10
iC4H10
357
357
357
357
357
357
357
357
357
357
357
357
357
357
357
357
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
5.176
C 5H 2
C 5H 3
C 5H 5
C 5H 6
357
357
357
357
5.176
5.176
5.176
5.176
C 6H
C 6H 2
C 6H 3
C 6H 6
C 6H 5
C6H5CH3
C 6H 5C 2H 3
C6H5CH2
C 6H 5C 2H
C10H8
357
357
357
464.8
464.8
495.3
546.2
495.3
535.6
630.4
5.176
5.176
5.176
5.29
5.29
5.68
6.00
5.68
5.72
6.18
7-23
Stanford University
Version 1.2
©Hai Wang
7.7.2 Multicomponent Properties
In section 7.4 we introduced the formula to calculate the mixture-averaged diffusivity Dk
from the binary diffusion coefficient Djk. A simple formula for mixture-averaged thermal
conductivity exists and is given by
⎛
⎞
1
1
K
⎜
⎟.
λ = ∑ k=1 X k λk + K
2⎜
∑ k=1 X k λk ⎟⎠
⎝
(7.75)
Again, both equations (7.21) and (7.75) are approximate. A rigorous treatment, known as
the multicomponent transport properties, may be found in a large number of texts and
papers* (also see, the attached document by Kee et al.). Computationally this treatment
should be used for numerical simulation of laminar premixed flames, along with a
description of the thermal diffusion of all species.
7.8 Flame Simulation and Flame Structures
The simulation of laminar premixed flames is customarily carried out using the Sandia
CHEMKIN and PREMIX suite of programs.** Here we shall discuss the structure of
several representative flames, computed using the PREMIX suite of codes.
Figure 7.11 shows the flame structure of the stoichiometric hydrogen-air flame at the
ambient pressure. Note that the onset of temperature rise from the room temperature to
about 2000 K occurs within a distance of ~1 mm, indicating that the flame is very thin. As
expected, the decreases in the reactant (H2 and O2) mole fractions are accompanied by the
rise in the product H2O mole fraction as well as the rise in the local temperature. The free
radical intermediates include mainly H•, O•, and OH•. The mole fractions of H• and O•
reach their respective peak values, they then decay somewhat slowly. The final values are, of
course, those defined by chemical equilibrium. For the OH• radical we see only a mild decay
in its concentration. Other intermediates, including HO2• and H2O2, are present in small
concentrations and never exceed the level of 1000 PPM. The calculated flame speed is 204
cm/s, which agrees almost exactly with the experimental value. This same agreement is seen
to extend into the entire range of equivalence ratios (Figure 7.8).
Figure 7.12 shows the structure of a stoichiometric methane-air flame, computed using the
GRI-Mech 3.0 (G. P. Smith, D. M. Golden, M. Frenklach, N. W. Moriarty, B. Eiteneer, M.
Goldenberg, C. T. Bowman, R. K. Hanson, S. Song, W. C. Gardiner, Jr., V. V. Lissianski,
and Z. Qin http://www.me.berkeley.edu/gri_mech/).
See, G. Dixon-Lewis, Proc. Roy. Soc. A. 307, 111-135 (1968); J. O. Hirschfelder, C. F. Curtiss and R. B.
Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, New York, 1954.
** R. J. Kee, J. F. Grcar, M. D. Smooke, J. A. Miller, A FORTRAN program for modeling steady
laminar one-dimensional premixed flames, Sandia Report 85-8240.
*
7-24
©Hai Wang
101
2000
Mole Fraction, Xk
1000
100
0
H2O
H2
10-1
O2
H
10-2
.
OH
.
O
10-3
0.00
Temperature (K)
Stanford University
Version 1.2
0.05
.
0.10
0.15
x (cm)
Figure 7.12. Flame structure of a stoichiometric hydrogen-air flame at the ambient pressure,
computed using the reaction model of Davis et al. (see, Figure 3.7).
7-25
©Hai Wang
101
2000
Mole Fraction, Xk
1000
100
0
O2
10-1
H 2O
CH 4
Temperature (K)
Stanford University
Version 1.2
CO 2
CO
H2
10-2
10-3
0.00
0.05
0.10
0.15
x (cm)
Figure 7.13. Flame structure of a stoichiometric methane-air flame at the ambient pressure,
computed using the GRI-Mech.
7-26