Environmental Engineering Reference
In-Depth Information
The solution of the second equation in (5.21) is
y
D
1.2, so
exp
1.19
T
0
E
a
AE
a
L
2
T
0
E
a
T
0
D
11.5 ,
Δ
T
D
.
(5.22)
From this, it follows that the connection between the specific powers of heat release
is
f
(
T
w
)
D
f
(
T
0
)exp
1.19
D
0.30
f
(
T
0
).
This gives the following connection between the parameters for heat release at the
walls at the threshold of thermal instability:
exp
L
2
AE
a
T
w
E
a
T
w
D
3.5 .
(5.23)
Although we referred to the Arrhenius law for the temperature dependence of the
rate of heat release, in actuality we used only the implication that this dependence
is strong. We can, therefore, rewrite the condition for the threshold of the thermal
instability in the form
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
D
L
2
df
(
T
w
)
dT
w
3.5 .
(5.24)
The criterion for a sharp peak in the power involved in the heat release is
T
w
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
df
(
T
w
)
dT
w
1 .
(5.25)
Relation (5.24) has a simple physical meaning. It relates the rate of the heat re-
lease process to the rate of heat transport. If their ratio exceeds a particular value of
the order of unity, then thermal instability develops.
5.1.7
Thermal Waves
The development of thermal instability can lead to the formation of a thermal wave.
This takes place if the energy in some internal degree of freedom is significantly in
excess of its equilibrium value. It can occur in a chemically active gas in combus-
tion processes, and the theory of propagation of thermal waves [20-23] used below
was developed for these processes. But the thermal explosion that is accompanied
by propagation of a thermal wave has a universal character and may be employed
also for problems involving ionized gases. Below we will be guided by a thermal
wave propagating in a nonequilibrium molecular gas where the vibration temper-
ature exceeds significantly the gas translation temperature, as occurs in molecular
gas lasers, in particular, CO
2
and CO gas lasers. In these cases the development of
thermal instability leads to a vibrational relaxation that is accompanied by equaliz-
ing of the translational and vibrational temperatures and proceeds in the form of a
propagating thermal wave. Below we will be guided by this problem.
