Environmental Engineering Reference
In-Depth Information
the above expression for the Rayleigh number, yielding
l
r
ν
Re
2
10
3
Ra
D
.
Since
l
1. Thus, at large values of the Rayleigh
number when several types of motion of gaseous flows are possible, the movement
of the gas is characterized by large values of the Reynolds number. As these values
increase, the gas motion tends toward disorder, and turbulent motion develops.
r
and
ν
,itfollowsthatRe
5.1.6
Thermal Explosion
We shall now consider the other type of instability of a motionless gas, which is
called thermal instability or thermal explosion. It occurs in a gas experiencing heat
transport by way of thermal conduction, where the heat release is determined by
processes (e.g., chemical) whose rate depends strongly on the temperature. There
is a limiting temperature where heat release is so rapid that thermal conductivity
processes cannot transport the heat released, and then an instability occurs, which
is the thermal explosion [13, 14]. As a result of this instability, the internal energy of
the system is transformed into heat, which leads to a new regime of heat transport.
We use the Zeldovich approach [15, 16] in the analysis of this problem.
We shall analyze this instability within the framework of the geometry of the
Rayleigh problem. Gas is located in a gap between two infinite plates with a dis-
tance
L
between them. The wall temperature is
T
w
.Wetakethe
z
-axis as perpen-
dicular to the walls, with
z
D
0 in the middle of the gap, so the coordinates of walls
are
z
L
/2. We introduce the specific power of heat release
f
(
T
)asthepower
per unit volume and use the Arrhenius law [17],
D˙
A
exp
,
E
a
T
f
(
T
)
D
(5.19)
for the temperature dependence of this value, where
E
a
is the activation energy of
the heat release process. This dependence of the rate of heat release is identical
to that for the chemical process and represents a strong temperature dependence
because
E
a
T
. This dependence is used widely for chemical processes [18, 19].
To find the temperature distribution inside a gap in the absence of the thermal
instability, we note that (4.41) for the transport of heat has the form
d
2
T
dz
2
C
f
(
T
)
D
0.
T
0
)/
T
0
,where
T
0
is the gas temperature
in the center of the gap, and obtain the equation
We introduce a new variable
X
D
E
a
(
T
d
2
X
dz
2
B
exp(
X
)
D
0,
