Environmental Engineering Reference
In-Depth Information
and this expression holds true for both a charged and a neutral test particle if the
mobility of a neutral atomic particle is taken as the limit
e
!
0. Here
T
is the gas
temperature in energy units, as usual,
μ
is the reduced mass of a test particle and
σ
(
g
) is the diffusion cross section for collision of a test
particle and a gas atom at relative velocity
g
, and brackets mean an averaging over
the Maxwell distribution function for gas atoms and test particles.
By analogy with (4.53) for the diffusion coefficient of a test atomic particle in
a gas, we have the following expression for the thermal conductivity coefficient
[11, 12]:
a gas atom or molecule,
2
5
p
π
T
D
2
p
m
.
(4.54)
32
σ
Here
m
is the mass of a gas atom or molecule, and the average cross section
σ
2
for
collision of gas atoms is given by
Z
1
D
μ
g
2
2
T
(2,2)
(
T
)
t
2
exp(
(2)
(
t
)
dt
,
σ
Ω
D
t
)
σ
t
,
2
0
Z
(1
(2)
(
t
)
cos
2
σ
D
#
)
d
σ
.
(4.55)
The viscosity coefficient
η
in the first Chapman-Enskog approximation is given
by [11, 12]
5
p
π
T
m
η
D
,
(4.56)
24
σ
2
where the average cross section
2
for collision of gas atoms is given by (4.55). As
is seen, we have the following relation between the coefficients of thermal conduc-
tivity and viscosity in the first Chapman-Enskog approximation:
σ
15
4
m
η
D
.
(4.57)
It should be noted that elastic collisions of atoms or molecules at thermal ener-
gies are described more or less by the hard-sphere model, where the interaction
potential of colliding particles is given by (2.10). Then the average cross sections of
collisions for the first Chapman-Enskog approximation are
2
3
σ
σ
D
σ
I
σ
D
0
,
(4.58)
0
2
R
0
and
R
0
is the hard-sphere radius. Correspondingly, (4.53), (4.54),
and (4.56) for the kinetic coefficients in the first Chapman-Enskog approximation
are simplified and take the form
where
σ
D
π
0
3
p
π
75
p
π
5
p
π
T
T
Tm
D
D
8
p
2
0
I D
0
p
m
I
η
D
.
(4.59)
μ
N
σ
64
σ
16
σ
0
