Environmental Engineering Reference
In-Depth Information
Figure 5.4
The temperature distribution in a gas in which a thermal wave propagates.
Figure 5.4 shows the temperature distribution in a gas upon propagation of a
thermal wave. Region 1 has the initial gas temperature. The wave has not yet
reached this region at the time represented in the figure. The temperature rise
observed in region 2 is due to heat transport from hotter regions. The temperature
in region 3 is close to the maximum. Processes that release heat occur in this re-
gion. We use the strong temperature dependence to establish the specific power of
heat release, which according to (5.19) has the form
f
(
T
m
)exp
α
T
)
,
E
a
T
m
.
f
(
T
)
D
(
T
m
α
D
(5.26)
Here
T
m
is the final gas temperature, determined by the internal energy of the gas.
Region 4 in Figure 5.4 is located after the passage of the thermal wave. Thermody-
namic equilibrium among the relevant degrees of freedom has been established in
this region.
To calculate the parameters of the thermal wave, we assume the usual connection
between spatial coordinates and the time dependence of propagating waves. That
is, the temperature is taken to have the functional dependence
T
ut
),
where
x
is the direction of wave propagation and
u
is the velocity of the thermal
wave. With this functional form, the heat balance equation (4.41) becomes
D
T
(
x
d
2
T
dx
2
u
dT
f
(
T
)
c
p
N
D
dx
C
C
0 .
(5.27)
Our goal is to determine the eigenvalue
u
of this equation. To accomplish this, we
employ the Zeldovich approximation method [15, 16], which uses a sharply peaked
temperature dependence for the rate of heat release. We introduce the function
Z
(
T
)
D
dT
/
dx
,sofor
T
0
<
T
<
T
m
we have
Z
(
T
)
0. Because of the equation
dT
dx
d
2
T
dx
2
d
dx
ZdZ
dT
D
D
,
equation (5.27) takes the form
ZdZ
dT
f
(
T
)
c
p
N
D
uZ
C
C
0 .
(5.28)
