Environmental Engineering Reference
In-Depth Information
D
is shown in Figure 5.6b, illustrating the distri-
bution of the gas temperature and number density of excited molecules along the
thermal wave for this case. Because at
x
The opposite limiting case
>
0 the rate of vibrational relaxation is low,
and at
x
0 excited molecules are absent, one can ignore the last term in (5.37). If
we assume the transition region to be small, (5.37) leads to the result
<
T
0
)exp
,
u
(
x
C
x
0
)
T
(
x
)
D
T
0
C
(
T
m
x
>
x
0
I
T
(
x
)
D
T
m
,
x
<
x
0
,
where
x
0
is the back boundary of the thermal wave. This value can be determined
from the condition that the positions of the centers for the gas temperature distri-
butions and the number density of molecules are coincident; that is, the areas of
the shaded regions in Figure 5.6b must be the same. This yields
x
0
D
/
u
and
T
m
1
1
e
T
0
e
T
(0)
D
T
r
D
C
,
(5.41)
where
e
is the base of Naperian logarithms. The wave velocity is determined by the
Zeldovich formula (5.35), where
f
(
T
)
D
Δ
ε
Nk
(
T
)and
N
is the step function.
This allows one to take
T
r
as the upper limit of integration in (5.35), leading to the
result
s
2
1
f
(
T
r
)
c
p
N
u
D
.
T
r
T
0
α
This formula, with (5.38) and (5.41), yields
r
2
e
e
r
T
r
p
E
a
(
T
m
u
D
,
(5.42)
1
τ
(
T
)
T
0
)
[
Nk
(
T
r
)]
1
and
E
a
/
T
r
.Since
where
τ
(
T
r
)
D
α
D
α
(
T
r
T
0
)
1, we conclude
that
r
k
(
T
)
u
.
In this case the wave velocity is slow compared with that for the case
D
be-
cause the vibrational relaxation process proceeds at lower temperatures and lasts
longer than in the case
D
D
. Summing up the above results, we point out that
the vibrational relaxation thermal wave is created by the processes of diffusion of
excited molecules in a gas, by thermal conductivity of the gas, and by vibrational
relaxation of excited molecules. Hence, the wave velocity depends on
D
and
D
and
on a typical time
τ
for vibrational relaxation.
5.1.9
Ozone Decomposition Through Thermal Waves
The process of propagation of a thermal wave results from competition between the
processes of heat release and heat transport, where the temperature dependence for