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
 
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