Environmental Engineering Reference
In-Depth Information
where
is the kinematic viscosity coefficient. This equation describes evo-
lution of a moving gas by accounting for the gas viscosity and the action of external
fields.
ν
D
η
/
4.1.4
Macroscopic Equation for Ion Motion in a Gas
We now derive the motion of ions in a gas subjected to an external electric field
E
by accounting for collisions of ions with gas atoms. The collision integral is given
by(3.8)inthiscase,andifthenumberdensityofionsissmallcomparedwiththe
number density of atoms, elastic collisions with atoms govern the character of the
motion of ions in a gas. We first use the tau approximation (3.5) for the collision
integral. Multiplying the kinetic equation (3.4) by the ion momentum
m
v
and inte-
grating it over ion velocities, we obtain the balance equation in the stationary case
in the form
m
w
i
τ
e
E
D
,
(4.16)
where
w
i
is the average ion velocity and
is the characteristic time between succes-
sive collisions of the ion with atoms. This time can be estimated as
τ
)
1
,
where
N
a
is the number density of atoms,
v
is a typical relative velocity of an ion-
atom collision, and
τ
(
N
a
v
σ
is a typical cross section for collisions with scattering into
large angles. The left-hand side of this equation is the force on the ion from the
electric field, and the right-hand side is the frictional force arising from collisions
of the ion with the gas atoms. We shall determine the frictional force below without
resorting to the tau approximation.
If we multiply the kinetic equation (3.4) by the ion momentum
m
v
and integrate
it over ion velocities
d
v
, we obtain the following balance equation by accounting
for expression (3.8) for the collision integral:
σ
Z
m
v
i
v
i
gd
e
E
N
i
D
σ
f
1
f
2
d
v
i
d
v
a
.
(4.17)
The quantities
v
i
and
v
a
are the initial velocities of the ion and atom, respectively,
of masses
m
and
m
a
,
N
i
and
N
a
are the number densities of ions and atoms,
g
is the relative velocity of ion-atom collision that is conserved in the collision, and
f
i
and
f
a
are the distribution functions for ions and atoms. From the principle of
detailed balance, which ensures invariance under time reversal in the evolution of
the system, it follows that
Z
Z
v
i
f
i
f
a
d
v
i
f
i
f
a
d
σ
d
v
i
d
v
a
D
σ
d
v
i
d
v
a
.
The invariance under time reversal (
t
d
v
i
d
v
a
must be examined. Let us
express the ion velocity
v
i
in terms of the relative ion-atom velocity
g
and the center-
of-mass velocity
V
by the relation
v
i
!
t
)of
d
σ
D
g
C
m
a
V
/(
m
C
m
a
). We find that
m
(
v
i
v
i
)
D
μ
(
g
g
0
). The relative velocity after collision has the form
g
0
D
g
cos
# C
k
g
sin
#
,
