Environmental Engineering Reference
In-Depth Information
employed for diffusion and thermal conductivity coefficients. The force acting per
unit area as a result of the momentum transport is
F
N
m
Δ
w
x
,where
N
v
v
is the particle flux and
m
w
x
is the difference in the mean momentum carried
by particles moving in opposite directions at a given point. Since particles reaching
this point without colliding are located from it at distances of the order of the mean
free path
Δ
λ
,wehave
m
Δ
w
m
λ
@
w
x
/
@
z
. Hence, the force acting per unit area is
z
. Comparing this with (4.28), and using (
T
/
m
)
1/2
F
N
m
λ
@
w
x
/
@
instead of
v
v
)
1
instead of
and (
N
σ
λ
, we obtain the estimate
p
mT
σ
η
(4.51)
for the viscosity coefficient
. The viscosity coefficient is found to be independent
of the particle number density. As was true for the thermal conductivity coefficient,
this independence comes from the compensation of opposite effects occurring with
the momentum transport. The number of momentum carriers is proportional to
the number density of atoms, and a typical transport distance is inversely propor-
tional to it. The effects offset each other.
To determine the force acting on a spherical particle of radius
R
moving in a gas
with velocity
w
, we assume that the particle radius is large compared with the mean
free path,
R
η
, and that the velocity
w
is not very high, so the resistive force
arises from viscosity effects. The total resistive force is proportional to the particle
area, so (4.28) gives
F
λ
Rw
. A more precise determination of the numerical
coefficient gives the Stokes formula for the frictional force that acts from the gas to
the spherical particles due to collisions of gas molecules [22, 23]:
η
F
D
6
πη
Rw
.
(4.52)
4.2.8
The Chapman-Enskog Approximation for Kinetic Coefficients of Gases
The kinetic coefficients of gases are determined by elastic scattering in collisions
of gas atoms or molecules, since the elastic cross section exceeds significantly the
cross section for inelastic collisions. The Chapman-Enskog approximation for ki-
netic coefficients accounts [24-26] for the simple connection between the kinetic
coefficients and the cross sections of elastic collisions of atoms or molecules and
is an expansion over a small numerical parameter. Therefore, we are restricted by
the first Chapman-Enskog approximation only. The expression for the ion mobil-
ity in the first Chapman-Enskog approximation was derived earlier and is given
by (4.21). From this, on the basis of the Einstein relation (4.37) we obtain the diffu-
sion coefficient
D
of a test particle in a gas in the first Chapman-Enskog approxi-
mation as [11, 12]
3
p
π
Z
g
2
2
T
T
8
N
p
2
1
2
D
μ
(1,1)
(
T
)
t
)
t
2
σ
(
t
)
dt
,
D
D
,
σ
Ω
D
exp(
t
,
(4.53)
μ σ
0
