Environmental Engineering Reference
In-Depth Information
A sequence of six representative snapshots that illustrate the dynamic behavior
of the convective flow observed inside the cell is shown in Fig.
4
. The velocity field
obtained with the PIV technique is superposed to the liquid region to give quantitative
information on the convective pattern. The most important feature of the flow is
that the motion of the solidification front has the effect of modifying the aspect
ratio of the volume where the fluid moves and therefore it has a definite influence
on the dynamics and convective pattern. The dynamics of the convective flow is
conveniently described in terms of the Rayleigh number (
Ra
) defined as
T
h
3
ʱʽ
gʲʔ
Ra
=
,
(1)
where
g
is the acceleration of gravity,
are respectively the coefficient of
thermal expansion, the thermal diffusivity and the kinematic viscosity of the liquid.
Since the thermal properties are functions of temperature, we have considered average
values. The temperature difference between the upper and lowermost regions of the
fluid, which in our case coincide with the solidification front, and the lower wall
of the cell is
ʲ
,
ʱ
and
ʽ
ʔ
T
. As shown in Fig.
2
, the distance between the upper and lower
boundaries of the liquid region in the cell is denoted by h. Notice that this distance
reduces in time as the solidification front advances and, as was mentioned in the
first paragraph of this section,
T
is also time dependent and therefore the Rayleigh
number is modified as the solidification progresses.
At the onset of the experiment, when the aspect ratio
A
ʔ
1, a single convec-
tive cell moving counterclockwise is observed. The average velocity is 0.66mm/s,
the Rayleigh number is
Ra
=
h/w
=
10
5
and the energy per unit mass of the system is
=
10
−
7
m
2
/s
2
. Then, the single cell evolves into two symmetric cells with fluid
descending in the central region and ascending near the vertical walls of the cell as
shown in Fig.
4
b, at t
4.4
×
=
3.5 min and
A
=
1 since no advance of the solidification
10
4
. The average velocity is 0.73mm/s and the
front can be noticed,
Ra
=
6
.
9
×
10
−
7
m
2
/s
2
. In later stages, the two cell configuration persists (see
Fig.
4
c, d), and then the patterns of convective motion in the cell becomes highly
irregular and undergo time dependent motions with time scales of the order of sec-
onds when the aspect ratio is in the range 0.6
energy is 5.3
×
0.5, as illustrated in Fig.
4
e.
At approximately 15.7 min after the onset of the observation when the aspect ratio
A
<
A
<
0.45, the liquid motion stops and the solidification proceeds in a stagnant fluid
for the rest of the observation. The critical Rayleigh number when the convective
motion stops is
Ra
c
∼
10
3
.
The shape of the solidification front at different times is shown in Fig.
5
.The
front grows slightly faster near the center but its curvature is small. This effect is
attributed to imperfect thermal insulation at the vertical walls of the cell. Observe
that this feature is consistent with the rotation direction of the double cells displayed
in Fig.
4
b where the relatively hot fluid ascends near the vertical walls. It was found
that at the scale of the cell, the geometry of the front is stable in the sense that no
preferential growth positions along the solid front were detected; the whole front
moves approximately at the same speed toward the lower wall.
=
3
.
2
×
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