Environmental Engineering Reference
In-Depth Information
molecules to be small compared with the initial density. We introduce the mean
energy
released in a single vibrational relaxation event. Then the difference
between the gas temperatures after
T
m
and before
T
0
the thermal relaxation wave
is
Δ
ε
N
0
Δ
ε
Nc
p
T
m
T
0
D
,
where
N
0
is the initial number density of excited molecules.
The heat balance equation (5.27) now has the form
u
dT
d
2
T
dx
2
Δ
ε
N
k
(
T
)
c
p
dx
C
D
0 .
(5.37)
The wave velocity can be obtained from the simultaneous analysis of (5.36) and
(5.37). The simplest case occurs when
D
. Then, both balance equations are
identical, and the relation between the gas temperature and the number density of
excited molecules is
D
N
Δ
ε
Nc
p
T
m
T
D
.
(5.38)
We have only one balance equation in this case. Comparing it with (5.27), we have
f
(
T
)/(
c
p
N
)
D
(
T
m
T
)
Nk
(
T
). On the basis of the Zeldovich formula we find the
wave velocity to be
s
2
T
m
E
a
(
T
m
u
D
,
(5.39)
T
0
)
τ
(
T
m
)
where
1/[
Nk
(
T
m
)] is a typical time for vibrational relaxation at tempera-
ture
T
m
. Because of the assumption
τ
(
T
m
)
D
1 and the dependence (5.26)
for the rate constant for vibrational relaxation, relation (5.39) with the assumptions
employed gives
α
(
T
m
T
0
)
r
u
.
τ
(
T
m
)
We shall now examine the propagation of a vibrational relaxation thermal wave
for limiting relations between
D
and
.Fig-
ure 5.6a shows the distribution of the number density of excited molecules
N
and of the gas temperature
T
along the wave. We note that the centers of these
two distributions coincide. This reflects the fact that the excited molecules that are
quenched at a given time introduce heat into the gas.
We can analyze the balance equation (5.36) for the number density of excited
molecules in a simple fashion by ignoring thermal conductivity processes. We ob-
tain the temperature distribution in the form of a step as shown in (Figure 5.6a).
At
x
.Weanalyzefirstthecase
D
>
0 the vibrational relaxation is weak, and the solution of (5.36) in this region