Jökull - 01.12.1973, Page 32

previously. Then (57) reduces to equation (58).
In (58) Ti has been replaced by the quantity
is the average, steady state conduction gradient
in the fluid.
The expression on the right of (58) represents
the critical Rayleigh number for an internally
heated fluid with two phases.
DISCUSSION
Inequality (58) shows that the critical Ray-
leigh number for a two-phase fluid is different
from that for a fluid with only one phase.
Basically there are two opposing effects. For a
normal phase transition, the latent heat tends
to promote stability by raising the critical
number whereas the density difference tends to
reduce stability by lowering the critical number.
These effects reverse their respective roles in
the case of anormal transitions. Inequality (58),
then, expresses quantitatively the phase transi-
tion effects which were mentioned previously.
In addition to the phase transition para-
meters, however, the physical constant , the
depdt h, and the ratio di/h influence the
critical condition. The effects of the phase
transitions are not very sensitive to the ratio
di/h. To simplify the discussion, the transition
will be fixed at the mid-plane of the fluid
layer. With di = h/2, (58) becomes,
3M 3Ap /j _ M \
4c^xh ^pi ^ctgvh \ c^hj
Defining the quantities (60)
3M M “ 4c^h (61)
p , _ 3AP Ae 2Pl2agYh (62)
the critical condition (60) is written
30 JÖKULL 23. ÁR
R >_______________________________ (63)
- l-RM + RAo (1-(4/3)Rm)
The expression RM represents the stabilizing
effect of the latent heat; R^g represents the
destabilizing influnce of the density difference
between the phases.
Fig. 7 shows the variation of Rc with RM for
different fixed values of R^f). It is observed
that for a given Raq> tl16 critical number in-
creases as RM increases. This is expected in view
of the stabilizing effect of the latent heat. Also,
for a fixed RM, the critical number decreases
as Rao increases. This reflects the fact that as
the density difference associated with the phase
change increases, the phase transition assumes
an increasingly important role in driving the
convection. An important feature of these
curves is that as RM approaches 0.75, the critical
number becomes insensitive to RAo > and for
Rm > 0.75, the critical number increases ex-
tremely rapidly. Consequently, a condition is
obtained on RM. In order for the critical
number to be finite, it is necessary that
hence
Rm<1
M
cPlh
<4/3
(64)
Fig. 7. Critical Rayleigh number Rc for con-
vection in a normal two-phase fluid as a func-
tion of Rm for various values R Aq •