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ʱ
where
k
ʱ
is the turbulent kinetic energy density for phase
and
k
2
ʱ
C
μʱ
ˁ
ʱ
μ
ʱ,
t
=
,
(10)
ʵ
ʱ
ʵ
ʱ
is the coefficient of the turbulent eddy viscosity, where
is the dissipation rate of
turbulence for phase
is a parameter that depends on the phase volume
fraction. In the actual simulations the turbulent eddy viscosity in relation (
9
)is
replaced by an effective dynamic viscosity, which is a sum of the dynamic and eddy
viscosities, i.e.,
ʱ
and
C
μʱ
k
in relation (
9
) and a sum is taken, as in expression (
5
), over all bubbles contained
in a unit cell. The transport equations for the turbulent kinetic energy
k
μ
ʱ,
eff
=
μ
ʱ
+
μ
ʱ,
t
. Note that for the dispersed phase
ʱ
=
b
,
and its
ʱ
dissipation rate
,aretakenas
those derived explicitly in the RNG procedure for a single phase (Yakhot and Smith
1992
; Pope
2000
).
The transport equations are solved using the
FLUENT 6.3
code for a cylindrical
water vessel of diameter 27.5 cm and height 40 cm, with its axis of symmetry aligned
along the
z
-axis. An eccentric nozzle exit of 6 cm diameter is placed at the bottom of
the vessel at one-third of its radius away from the wall, through which air is allowed
to flow at a constant velocity. At the beginning of the calculations, the water volume
fraction
ʵ
ʱ
as well as the values of all constants, including
C
μʱ
ˆ
b
is set to zero. No-slip boundary
conditions for the fluid velocities apply along the wall of the vessel and the water
free surface is assumed to remain flat. However, air bubbles reaching the free surface
are allowed to flow out. To do so a sink term is added to Eq. (
1
) for the dispersed
phase for those control volumes at the free surface. A tetrahedral mesh covering the
full cylindrical domain was created with the help of the commercial software
Gambit
2.0
and trial calculations were conducted to determine convergence of the solutions.
True convergence is guaranteed using a tetrahedral mesh with 333,465 elements.
ˆ
w
=
1, while the air volume fraction
3 Results and Discussion
Most industrial gas-stirred ladles have their porous plugs located off-centre because
it is believed that this configuration produces a better stirring effect than when the
porous plug is placed at the centre. Therefore, we simulate the injection of air into a
water vessel when the plug position is one-third of the vessel radius away from the
wall for two different constant airflow rates (i.e.,
Q
10
−
5
m
3
s
−
1
) in order to investigate the effects on the flow field. In the present model, the
bubbles are assumed to be spherical with a uniform size distribution and released
from the nozzle exit with constant frequency.
For
Q
10
−
6
and 1
=
9
.
01
×
.
03
×
0.009 l s
−
1
), Fig.
1
shows the resulting stable flow
pattern. The central axis of the plume is evidenced by the nearly vertical red pathlines,
while the yellow ones closely coincide with its lateral borders. In the upper portion of
the ladle, just above the plume region, there is no recirculation and the flow pathlines
10
−
6
m
3
s
−
1
(
=
9
.
01
×
≈
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