Environmental Engineering Reference
In-Depth Information
4.3.3
Thermal Conductivity of Electrons in an Ionized Gas
Because of their small mass, electrons are effective carriers of heat, and their trans-
port can give a contribution to the thermal conductivity of a weakly ionized gas.
Since the thermal conductivity of a gas is determined by different carriers, the ther-
mal conductivity of an ionized gas is the sum of the thermal conductivity due to
atoms and electrons independently. Below we evaluate the thermal conductivity co-
efficient due to electrons in an ionized gas with a low degree of ionization, where
the number density of electrons
N
e
is small compared with that of atoms
N
a
,that
is,
N
e
N
a
.Weconsidertheregimeofahighelectronnumberdensityforelec-
tron drift in an external electric field, and the criterion to be fulfilled is the inverse
of criterion (3.30). In this regime the Maxwell distribution function for electron
velocities
(
v
) is realized, and the electron temperature
T
e
is the parameter of this
distribution.
The gradient of the electron temperature creates a heat flux
q
e
,andtheelectron
thermal conductivity
'
e
is defined by the equation
q
e
D
r
T
e
.
(4.69)
e
As usual, we assume the electron temperature gradient to be relatively small, so
criterion (4.24) for the mean free path of electrons in a gas
is fulfilled, and
L
in
this criterion is a typical distance over which the electron temperature
T
e
varies
remarkably.
To determine the electron thermal conductivity, we represent the velocity distri-
bution function for electrons in a standard way similar to (3.10):
λ
f
(
v
)
D '
(
)
C
(
v
r
ln
T
e
)
f
1
(
).
v
v
Assuming that variation of the electron momentum is determined by electron-
atom collisions, we obtain the kinetic equation in the form
v
r
f
D
I
ea
(
f
).
We assume that the spatial dependence of the electron distribution function is due
to the gradient of the electron temperature, and the equation of state for the elec-
tron gas is given by (4.11) for the electron pressure
p
e
N
e
T
e
, which is constant
in space. On the basis of (3.12) for the collision integral for the nonsymmetric part
of the electron distribution function, we reduce the kinetic equation to the form
D
(
v
)
m
e
v
2
2
T
e
5
2
'
v
r
T
e
D
ν
f
1
.
For the nonsymmetric part of the distribution function this gives
m
e
v
.
2
2
T
e
D
'
(
)
5
2
v
f
1
(
v
)
ν
