Geoscience Reference
In-Depth Information
The latter circumstance is not convenient, and we eliminate n.a/ by the following
trick. We find the product n.a/ from the condition that the mass flux of the carrier
gas molecules is equal to zero. This condition gives:
n.a/
D
˛n
1
(3.196)
where
D
p
T
1
=T
Substituting Eq.
3.196
into Eq.
3.195
allows for presenting the distribution f
s
in
the form
f
s
D
f
.0/
s
C
f
.1
s
;
where
D
n
1
M
1
L
2
.r/
L
2
:
f
.0/
s
(3.197)
This function describes the unperturbed distribution, that is, the carrier gas
molecules distributed in the empty space (no particle or a particle with ˛
D
0).
The function
D
˛n
1
.M
M
1
/
L
2
.r/
L
2
L
2
.a/
L
2
ı
s;1
f
.1/
s
(3.198)
describes the perturbation generated by the heated particle. Here ı
ik
is the Kroneker
delta and .x/is the Heaviside step function.
The contribution from f
.1/
to the mass flux should be zero in the steady state.
This condition is seen to be fulfilled. The function f
.1/
contains only outgoing
s
molecular trajectories.
Now let us find the free molecule heat transfer efficiency. Equation
3.195
gives
Q
D
2˛a
2
v
1
kn
1
.T
T
1
/
(3.199)
and
fm
.a/
D
2˛a
2
v
1
kn
1
: (3.200)
Here
v
1
D
p
8kT
1
=m is the thermal velocity of the carrier gas molecules.
To calculate the
q
-and
n
profiles we use the equation
L
2
.a/
Z
dL
2
p
L
2
.r/
L
2
D
2L.r/D.r/;
(3.201)
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