Environmental Engineering Reference
In-Depth Information
We now determine the heat flux due to electrons:
Z
m
e
v
Z
m
e
v
2
2
2
x
q
e
D
v
x
f
(
v
)
d
v
D
r
ln
T
e
f
1
(
v
)
d
v
.
v
2
2
Accounting for the connection between the symmetric and nonsymmetric parts of
the distribution function and (4.69) for the heat flux due to electrons, we obtain for
the thermal conductivity coefficient of electrons
N
e
v
m
e
v
,
2
m
e
v
2
2
2
T
e
5
2
D
(4.70)
e
3
ν
2
T
e
where angle brackets mean averaging over the Maxwell electron distribution func-
tion.
Let us consider the limiting cases. We take the dependence
n
ν
v
for the
0
z
n
/2
,where
z
2
/(2
T
e
).
rate of electron-atom collisions, that is,
ν
(
)
D
ν
D
m
e
v
v
Then (4.70) gives
7
.
0
m
e
1
2
4
3
p
π
T
e
N
e
ν
n
n
D
Γ
(4.71)
e
2
ν
D
In particular, if
const, this formula gives
5
T
e
N
e
2
D
.
e
ν
0
m
e
If
n
D
1, that is,
ν
D
v
/
λ
(
λ
is the mean free path), we have from (4.71)
s
2
T
e
m
e
2
3
p
π
e
D
N
e
λ
.
To determine the contribution of the electron thermal conductivity to the total
thermal conductivity coefficient, it is necessary to connect the gradients of the elec-
tron
T
e
and atomic
T
temperatures. Let us consider the case when the difference be-
tween the electron and gas temperatures is determined by an external electric field,
and the connection between the electron and gas temperatures is given by (3.36).
If
n
,thisformulagives
ν
v
r
T
r
T
e
D
.
nT
T
e
1
C
n
In particular, in the case
T
e
T
, the total thermal conductivity coefficient is
e
r
T
e
C
e
D
C
T
D
n
,
(4.72)
a
a
r
1
C
where
a
is the thermal conductivity coefficient of the atomic gas. If we use (4.40)
as an estimation for the gas thermal conductivity coefficient, replacing the electron
parameters in this formula by the atom parameters, we obtain that the electron
thermal conductivity can contribute to the total value at low electron number den-
sities
N
e
<
N
a
because of the small electron mass and the high electron tempera-
ture.
