Environmental Engineering Reference
In-Depth Information
ʵ
Re-Normalization-Group (RNG)
model for the governing equations of
turbulence (Yakhot et al. 1992 ; Yakhot and Smith 1992 ).
k -
2 Two-Phase Fluid Flow Model
The gas-stirred ladle is essentially a bubble-driven recirculation flow system. When
the air is released from the porous plug, the flow is governed by the inertial force of
the injected gas and large bubbles are formed at a small height from the nozzle exit
( primary or momentum region ). As the gas loses its kinetic energy, the large bubbles
disintegrate into small ones ( transition region ) and owing to the density difference
between air and water, they will float to the top until reaching the water surface
( bubble region ). When air bubbles float, they induce the water to flow and form a
recirculation region outside the plume-shaped, two-phase region. In the upper part of
the recirculation region the water flows towards the vessel walls. It then flows down
along them and finally gets the ladle bottom. Therefore, the system can be divided
into two zones: the plume zone, where air bubbles (dispersed phase) and water
(continuous phase) coexist, and the water zone, where recirculation takes place.
If we assume isothermal conditions, the equations governing this two-phase flow
in Eulerian form are: the mass conservation equation
( ˆ ʱ ˁ ʱ )
+∇· ( ˆ ʱ ˁ ʱ v ʱ ) =
,
0
(1)
t
and the momentum conservation equation
∂ (ˆ ʱ ˁ ʱ
v
ʱ )
+∇· ʱ ˁ ʱ
v
v
ʱ ) =− ˆ ʱ
p
+∇·
T
ʱ + ˆ ʱ ˁ ʱ
g
ʱ
t
−∇· ʱ ˁ ʱ ʴ
v
ʱ ʴ
v
ʱ ) +
F
ʱ ,
(2)
for each phase
w and b for the water and bubbly phase, respectively, and
where the phase pressures have been taken to be equal, i.e., p
ʱ
, where
ʱ =
p . This assumption
is considered to be valid except in the case of expanding bubbles (Drew 1983 ). Each
phase has its own intrinsic mass density
ʱ =
ˁ ʱ
, velocity v
, volume fraction
ˆ ʱ
, and
ʱ
viscous stress tensor
2
3 ˆ ʱ μ ʱ ( ∇·
v T
ʱ
T ʱ = ˆ ʱ μ ʱ
v ʱ +∇
v ʱ )
I
,
(3)
where
, I is the identity tensor, and the super-
script T means transposition. In addition to Eqs. ( 1 ) and ( 2 ) we have the following
constraints
μ ʱ
is the dynamic viscosity of phase
ʱ
ˆ ʱ =
1
=
ˆ ʱ ˁ ʱ ,
(4)
ʱ
ʱ
for the volume fractions and the density of the mixture.
 
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