Environmental Engineering Reference
In-Depth Information
The third term on the right-hand side of Eq. (
2
) is the gravitational acceleration
force directed along the negative
z
-axis, while the fourth and fifth terms are the
Reynolds stress (accounting for turbulent effects, where
ʴ
v
ʱ
=
v
l
,ʱ
−
v
ʱ
is the
fluctuating component of the velocity and
v
l
,ʱ
is the local instant velocity of phase
ʱ
) and a force per unit volume due to phase interactions which cause the transfer of
momentum between relatively moving phases. Here this term is given by the sum
of three additional forces: the drag force, which represents the force on a bubble
due to its velocity relative to the water, the lift force, which accounts for the effects
of velocity gradients in the water flow field on the bubbles, and the virtual mass
force due to the inertia of the water mass on the accelerating bubbles. We assume a
steady-state drag on the bubbles so that there is no acceleration of the relative velocity
between the bubbles and the conveying fluid. This force is given by (Mazumdar and
Guthrie
1995
; Crowe et al.
1998
)
N
Re
k
v
w
−
v
b
,
k
ˆ
b
,
k
,
C
D
μ
w
D
b
,
k
3
4
F
D
,
w
=−
F
D
,
b
=−
(5)
k
=
1
where
C
D
(
6mm is the mean
diameter of the spherical bubbles, Re
k
is the disperse phase relative Reynolds number
=
0
.
44 for Re
>
1,000) is the drag coefficient,
D
b
,
k
=
Re
k
=
ˁ
w
D
b
,
k
μ
w
|
v
w
−
v
b
,
k
|
,
(6)
and the sum is taken over the total number
N
of bubbles contained in a unit cell (or
control volume). The lift force is given by (Drew and Lahey
1987
)
F
L
,
w
=−
F
L
,
b
=−
0
.
5
ˆ
b
ˁ
w
(
v
w
−
v
b
)
×
(
∇×
v
w
) ,
(7)
while the virtual mass effects of the bath on the bubble column are given by (Drew
and Lahey
1987
)
ˆ
b
ˁ
w
d
v
w
d
v
b
dt
F
V
,
w
=−
F
V
,
b
=
0
.
5
dt
−
,
(8)
where
d
/
dt
denotes the phase material time derivative. We note that these forces act
as sinks for the continuous (water) phase and as sources for the dispersed (bubbles)
phase.
Turbulence is modelled using the Re-Normalization Group (RNG)
k
-
model
(Yakhot et al.
1992
; Yakhot and Smith
1992
). This method differs from the standard
k
-
ʵ
model in that it accounts for smaller scales of turbulence by means of a renor-
malization of the Navier-Stokes equations, thereby allowing for turbulence at high
Reynolds numbers. The Reynolds stresses in Eq. (
2
) are defined using Boussinesq's
assumption
ʵ
v
ʱ
=
μ
ʱ,
t
2
3
ˁ
ʱ
k
ʱ
I
v
T
ʱ
−
ˁ
ʱ
ʴ
v
ʱ
ʴ
∇
v
ʱ
+∇
−
,
(9)
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