Environmental Engineering Reference
In-Depth Information
where
f
0
is the equilibrium distribution function. If the equilibrium state of a gas
is disturbed, and the distribution function at the initial time is
f
(0), the subsequent
evolution of this gas within the framework of the tau approximation is described
by the following kinetic equation:
df
dt
D
f
f
0
.
τ
Its solution has the form
f
0
]exp
t
τ
f
D
f
0
C
[
f
(0)
.
As is seen, the relaxation time
is a parameter of the collision integral for the tau
approximation. It can be estimated as
τ
σ
v
)
1
according to the definition of
the relaxation time, where
v
is a typical collision velocity and
τ
(
N
σ
is a typical cross
section of elastic scattering of atoms.
We now give the expression for the collision integral if it is determined by elastic
collision of a test atom with gas atoms. Let us introduce the specific rate constant
for elastic collision of two atoms
W
(
v
1
,
v
2
!
v
1
,
v
2
), so
Wd
v
1
d
v
2
is the probability
per unit time and per unit volume for collision of two atoms with velocities
v
1
and
v
2
if their final velocities are in an interval from
v
1
to
v
1
C
d
v
1
and from
v
2
to
v
2
C
d
v
2
, respectively. Then, by definition, the collision integral is
Z
f
1
f
2
W
0
f
1
f
2
W
d
v
1
d
v
2
d
v
2
.
I
col
(
f
)
D
f
1
D
f
(
v
1
),
f
2
D
f
(
v
2
),
whereweusethenotation
f
1
D
f
(
v
1
),
f
2
D
f
(
v
2
),
W
(
v
1
,
v
2
!
v
1
,
v
2
).
We now analyze the properties of the specific rate constant
W
for elastic collisions
of atoms. The principle of detailed balance, which corresponds to time reversal
t
!
v
1
,
v
2
), and
W
0
D
W
D
W
(
v
1
,
v
2
W
0
. Indeed, on time reversal, the atom ensemble develops in
the inverse direction with the same atom trajectories, which corresponds to change
v
1
!
t
,gives
W
D
$
v
1
,
v
2
$
v
2
. This allows us to represent the collision integral in the form
Z
W
f
1
f
2
f
1
f
2
d
v
1
d
v
2
d
v
2
.
I
col
(
f
)
D
(3.6)
The specific rate constant
W
has another property if atoms 1 and 2 are identical.
Then replacing atoms gives
!
v
1
,
v
2
)
!
v
2
,
v
1
).
W
(
v
1
,
v
2
D
W
(
v
2
,
v
1
(3.7)
Let us replace the specific rate constant in the expression (3.6) for the collision
integral by the cross section of elastic collision of atoms. By definition, the elastic
scattering cross section is the ratio of the number of scattering events per unit time
to the flux of incident particles. Therefore, the differential cross section for elastic
scattering of atoms is given by
f
1
f
2
Wd
v
1
d
v
2
d
v
1
d
v
2
j
v
1
Wd
v
1
d
v
2
j
v
1
d
σ
D
f
1
d
v
1
f
2
d
v
2
D
.
v
2
j
v
2
j
