Environmental Engineering Reference
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ʔ
smaller since
T
is decreasing but the aspect ratio remains constant. Then, two effects
contribute to the reduction of the Rayleigh number, the characteristic distance h gets
smaller as time progresses and the temperature gradient is also smaller. The critical
Rayleigh number when the motion of the liquid stops is indicated with the second
vertical line. Although the solidification front is not a horizontal line as would be
required by a one-dimensional model, a simple one dimensional heat transfer balance
at the position where phase change takes place can be proposed, considering also that
the liquid is motionless. Upon assuming that the heat released at the solidification
front is conducted away by the solid phase, the velocity of the front is proportional to
the temperature difference at the front position and the temperature at the boundary
condition in the ice divided by the distance that separates them. i.e.
k
T
f
−
T
C
d
dt
−
=
Lv
f
=−
L
(2)
y
where
k
is the heat conductivity of ice,
T
f
is the temperature at the solidification
front, the
is the enthalpy of phase change and
y
is the position of the front which is
a function of time. The previous expression can be integrated considering that
y
=
L
0
=
at
t
0. The result indicates that the position of the front is given by:
L
t
1
/
2
y(
t
)
=
C
where
C
=
(3)
k
(
T
f
−
T
C
)
Even though the front is not a horizontal line but has a small curvature, the simple
model indicates the correct value of the exponent of time as compared with the
observation once the velocity of the liquid has become small enough at time
t
>
15.7 min. This feature is illustrated in the inset in Fig.
6
. Although the motion of the
solidification front has a major influence on the dynamics of the convective motion,
under our experimental conditions, the motion of the fluid does not greatly modify
the shape of the liquid-solid interface.
The intensity of the motion of the liquid is obtained by calculating the L
2
norm
of the flow defined by:
1
V
o
L
2
2
2
=
V
o
(u
+
v
)
dV
o
,
(4)
where
V
o
(
is the volume occupied by the liquid and is a function of time. The
velocity components in the
x
and
y
directions are
u
and
t
)
respectively. Under the
approximation of constant density, L
2
corresponds to twice the kinetic energy of
the system. As can be observed from Fig.
7
, the convective motion starts when the
unstable temperature gradient is established in the cell and the velocity increases at
t
v
3 min due to the increase in the temperature gradient and the concurrent condition
of constant aspect ratio of the liquid volume (see Fig.
3
). At approximately t
=
3min,
the solidification front starts moving downwards shortening the volume available to
the liquid and the kinetic energy of the system reduces monotonically until the fluid
=
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