Geoscience Reference
In-Depth Information
The steady-state heat flux
Q
(
a
) from the particle of radius
a
can always be
written as
Q
.a/
D
.a/.T
T
1
/:
(3.13)
The heat flux is directed outward from the particle. The proportionality coeffi-
cient .a/ is referred to as the heat transfer efficiency of the particle. Our goal is to
find .a/.
We also introduce the generalized efficiency .a;R/defined as follows:
Q
.a;R;T
R
/
D
.a;R/.T
T
R
/;
(3.14)
where q
R
is the temperature at a distance (yet arbitrary)
R
from the particle center.
Let us assume for a moment that we know the exact temperature profile T
exact
.r/
corresponding to the heat flux
Q
. Then, using Eq.
3.14
we can express
Q
(
a
)interms
of T
exact
as follows:
Q
.a/
D
.a;R/.T
T
exact
.R//:
(3.15)
Now let us choose
R
sufficiently large for the macroscopic approximation (the
thermoconductivity equation) to reproduce the exact temperature profile:
T
exact
R
D
T
Q
.R/
D
T
1
C
Q
=.4ƒR/;
(3.16)
with q
Q
.r/ being the steady-state
q
-profile corresponding to a given total heat flux
Q
and ƒ being the thermoconductivity of the carrier gas. Combining this with
Eq.
3.15
gives
.a/
D
.a;R/
T
T
1
Q
:
.a/
4ƒR
Q
(3.17)
We can solve this equation with respect to
Q
.a/,useEq.
3.14
,andfind.a/:
.a;R/
1
C
.a/
D
:
(3.18)
.a;R/
4ƒR
There is not a chance to find .a;R/exactly. We again call upon two approxi-
mations:
We approximate .a;R/by its free molecule expression:
.a;R/
fm
.a;R/:
(3.19)
Wedefine
R
from the condition
d
r
n
fm
.r/
j
rDR
D
d
r
n
Q
.r/
j
rDR
;
(3.20)
