Geoscience Reference
In-Depth Information
Substituting
Eqs.
3.5
and
3.7
into
Eq.
3.4
gives
the
equation J.a/
D
˛.a;R/n
.J/
.R/,or
J.a/
D
˛.a;R/e
ˇU.R/
0
1
Z
e
ˇU.r
0
/
dr
0
r
0
2
J.a/
4D
@
n
1
A
:
(3.8)
R
We can solve this equation with respect to J.a/ and find ˛.a/:
˛.a;R/e
ˇU.R/
˛.a/
D
:
(3.9)
R
˛
.
a;R
/
e
ˇU.R/
4D
e
ˇU.r
0
/
dr
0
r
0
2
1
C
Equation
3.9
is exact. We, however, know neither ˛.a;R/ nor
R
. It is pertinent
to notice here that if we have some grounds to neglect unity on the right-hand side
of Eq.
3.10
; then, ˛.a/ depends only on the separation distance
R
:
4D
˛.a/
D
:
(3.10)
R
e
ˇU.r
0
/
dr
0
r
0
2
There is no way to find ˛.a;R/ exactly. We thus call upon two approximations:
We approximate ˛.a;R/ by its free molecule expression
˛.a;R/
˛
fm
.a;R/
(3.11)
WedefineR from the condition
d
r
n
fm
.r/
j
rDR
D
d
r
n
.J.a//
.r/
j
rDR
;
(3.12)
where n
fm
.r/ is the ion concentration profile found in the free molecule regime for
a<r<R. The distance
R
separates the zones of free molecule and continuous
regimes.
3.2.2
Flux-Matching for Enthalpy Transport
Let us consider a spherical particle of radius
a
heated to temperature T
suspended
in the carrier gas with temperature T
1
<T
. In what follows we operate with the
mean energy per a particle
q
and the flux of this energy
rather than with the
temperature and temperature flux. The reasons for this are explained later, when we
define these values in terms of the distribution function.
Q
