Geoscience Reference
In-Depth Information
3.7.2
Free Molecule Solution
As we already saw, in spherically symmetrical systems the set of variables r;E;L,
is of extreme convenience, with L being the angular momentum of the carrier gas
molecule. In these variables the Boltzmann equation takes an especially simple
form:
@f
s
@r
D
0;
v
r
(3.193)
where
v
r
D
.mr/
1
p
L
2
.r/
L
2
is the radial ion velocity, and L
2
.r/
D
2mEr,
s
D˙
1 is an auxiliary variable defining the direction of ion motion along the radial
coordinate (s
D
1 corresponds to the direction toward the particle).
In the free molecule zone we operate with the mean kinetic energy q.r/,whichis
proportional to the temperature of the carrier gas. Now we give the exact definition
of the values q.r/, the energy flux
Q
.r/, and the number density n.r/:
Z
dE
Z
m
2
r
X
s
dL
2
p
L
2
.r/
L
2
f
s
.r;E;L/
n.r/
D
Z
EdE
Z
m
2
r
X
s
dL
2
p
L
2
.r/
L
2
f
s
.r;E;L/
q.r/
D
(3.194)
s
Z
EdE
Z
dL
2
f
s
.r;E;L/
m
3
X
s
4
2
Q.r/
D
Here
m
is the mass of a carrier gas molecule.
The boundary condition
f
1
.a;E;L/
D
.1
˛/f
1
.a;E;L/
C
M
.E/n.a/
L
2
.a/
L
2
: (3.195)
couples f
1
and f
1
on the particle surface. Here, n.a/ isgivenbyEq.
3.194
at
r
D
a, ˛ is the energy accommodation coefficient, and the asterisk (*) refers to the
particle surface:
M.E/
D
.m=2kT/
3=2
e
E=kT
is the Boltzmann distribution normalized to unity, is a factor providing the equality
of the inward and outward molecular fluxes (will be determined later on), and .x/
is the Heaviside step function. This boundary condition claims that part 1
˛ of
molecules experiences the mirror reflection from the particle surface whereas part
˛ sticks to the particle surface, acquires its temperature T
, and then isotropically
leaves the particle surface. Two points should be emphasized:
1. This boundary condition is
linear
in f .
2. The function f
1
appears on the right-hand side of the boundary condition.
