Environmental Engineering Reference
In-Depth Information
Using this character of particle evolution, one can represent the equation for the
particle distribution function, the kinetic equation, in the form
df
dt
D
I
col
(
f
),
(3.3)
where the left side of this equation describes evolution of a free test particle, and
I
col
(
f
) is the so-called collision integral, which takes into account the variation of
the number of particles in a given state as a result of pairwise collisions.
The left-hand side of the kinetic equation that describes the motion of particles
in external fields is
df
dt
D
f
(
v
C
d
v
,
J
,
r
C
d
r
,
t
C
dt
)
f
(
v
,
J
,
r
,
t
)
.
dt
For a free test particle that does not collide with surrounding gas particles, we have
the motion equation
d
v
/
dt
D
F
/
m
,where
F
is the force that acts on the test
particle from external fields,
m
is the particle mass, and the particle velocity is
v
D
d
r
/
dt
.Thus,wehave
df
dt
D
@
f
t
C
v
@
@
r
C
m
@
f
f
@
v
,
@
and the kinetic equation (3.3) takes the form
@
f
t
C
v
@
f
@
r
C
m
@
f
@
v
D
I
col
(
f
).
(3.4)
@
This is the Boltzmann kinetic equation [1] that describes the evolution of a particle
system.
We note that the kinetics of an ensemble of identical particles based on the ki-
netic equation (3.4) uses other principles compared with classical hydrodynamics
and thermodynamics of gases, where the average parameters of particles are taken
for each spatial point and time. The concept of the distribution function for gas
particles for gas dynamics was developed by Maxwell [2-4] and Boltzmann [5-9]
and led to the kinetic equation (3.4) [1] that describes gas dynamics through evolu-
tion of the distribution function for gas atoms or molecules. Note that the kinetic
description of the evolution of an ensemble of identical particles is suitable for gas
andliquidsystems[10-14].
3.1.2
Collision Integral for Gas Atoms
The collision integral contained in the kinetic equation characterizes the evolution
of the system as a result of pairwise collisions of atomic particles. The collision
integral is responsible for gas relaxation when this gas is moved out of equilibrium.
The simplest model for the collision integral corresponds to the tau approximation,
when the collision integral is approximated by the expression
f
f
0
I
col
(
f
)
D
,
(3.5)
τ
