Jökull - 01.12.1973, Page 26

Recalling that L = 2h, the critical condition
for convection becomes
. X + T
A-------
4r
(23)
agfih4 72hL'
V‘l T2
(20)
The ratio IiI//t2 is simply a numerical factor,
hence (20) is of the same form as the results
obtained by Jeffreys (1928).
Since the strip model leaves the aspect ratio
of the convection cell indeterminate, we must
choose \ and r on physical grounds. Intui-
tively it appears fairly evident that the lialf-
cell sketched in Fig. 2 will be nearly rectangular
and that X — 2h is a reasonable assumption.
This choice would also coincide witli the re-
sults of Jeffreys (1928) for the present case.
Inserting X = 2h and substituting for I/ from
(10), we can write (20)
R =
72(4-3 Q
i2
(21)
where £ = r/h. Making the physically reason-
able assumption that £ = 0.35 (21) becomes
Since the core is fairly small, tliis simplification
should not result in serious error. The lower
boundary of the cell is in contact with a therm-
ally insulating slab, hence the temperature gra-
dient is continuous at all points in the strip.
Assuming constant conductivity, we must solve
d2T dT
K Æ oe" *r = _<>A- <24>
with the boundary conditions
T(0) = T(L) = 0
For As = constant, the temperature T(x) is
R_ jío/Jh^ 1750
i/a
(22)
This result is quite close to the result of Jeffreys
(1928) of R = 1708, even though we have
neglected all horizontal heat transfer. On the
present model, the effect of horizontal heat
transfer would simply be to modify the value
of £■
APPLICATION TO AN
INTERNALLY HEATED FLUID
In order to determine the temperature distri-
bution in the strip model treatment, we place
all the heat sources in the cell uniformly
throughout the vertical branches. The specific
heat production in the vertical segments As
is adjusted so that the total heat production in
the cell is unchanged. Thus
ASVS = AV
where Vs is the volume of the vertical segments.
A and V are the specific production and volume
of the cell respectively, or
Inserting (25) into (9) we obtain for the total
head
H =
p2«gAsL3
4K ^ b
(26)
where
g(b) = 1/b (1 — 4/b tanh (b/4)). (26')
Substituting (26) into (8) and multiplying by
L/a yields
where now
p2AsagLíT2
N =-------------
KíjaL'
(27)
(28)
We seek the condition for which (27) has a
solution other than u = 0. For small b
g(b)
(29)
24 JÖKULL 23. ÁR