Environmental Engineering Reference
In-Depth Information
T
w
)/
T
w
into (5.130),
where
r
0
isthetuberadiusand
T
w
is the wall temperature. Then (5.130) can be
writtenintheform
d
dz
We introduce the new variables
z
D
2
/
r
0
and
θ
D
(
T
z
d
dz
C
A
exp(
b
θ
)
D
0 ,
(5.131)
where
A
D
r
0
p
(
T
w
)/[4
T
w
(
T
w
)],
b
D
T
w
d
ln
p
(
T
w
)/
dT
w
, and in this case
1. Because of the strong dependence of
p
on
T
, we can assume that the
thermal conductivity coefficient
b
is independent of the temperature. The solution
of (5.131) is given by the Fock formula:
1
b
ln
2
γ
θ
D
z
)
2
,
(5.132)
Ab
(1
C
γ
where
is a parameter to be determined by boundary conditions. One of the
boundary conditions of (5.131) is
γ
θ
(0)
D
0, following from the definition of this
function. This gives the relation for
γ
that
)
2
.
γ
D
Ab
(1
C
γ
2
(5.133)
Since
1/2. If this condition is not satisfied, then (5.131) has
no real solution. This means that thermal instability arises because heat extraction
is not compensated for by heat release. The instability then leads to a different
discharge regime where the current occurs in a narrow region near the tube center.
The threshold for this instability corresponds to
γ
is real, we have
Ab
γ
D
1and
Ab
D
1/2, that is, it
has its onset when
p
(
T
0
)
D
4
p
(
T
w
), where
T
0
is the temperature at the axis. The
instability threshold at
Ab
D
1/2 corresponds to
dp
(
T
)
dT
2
(
T
w
)
r
0
T
w
D
.
(5.134)
If this instability leads to contraction of the plasma, these relations make it possi-
ble to estimate the radius
0
for the new contracted regime of the plasma. Formu-
la (5.134) then gives
(
T
0
)
dp
(
T
)
dT
1
2
0
,
(5.135)
T
D
T
0
where
T
0
is the temperature at the axis.
5.5.3
Ionization Wave in a Photoresonant Plasma
In considering a photoresonant plasma in Section 3.3.9, we found two regimes
of plasma to exist. In the regime of low intensities of incident resonant radiation
being absorbed in a plasma thickness of approximately 1/
k
0
(for definiteness, we
assume the frequency of incident radiation is close to the center of the spectral line
