Environmental Engineering Reference
In-Depth Information
Electrolyte
flow
z
y
Electrodes
Permanent
magnets
x
Fig. 1
Sketch of the electrolytic flow past a pair of magnetic obstacles side by side. See details in
the text
L
is much smaller than the distance between lateral walls, are placed beneath the
bottom wall of the container with an orientation such that resulting Lorentz forces
oppose the oncoming flow and generate vorticity. Figure
1
shows a sketch of the
problem under consideration. Since the thickness of the fluid layer is assumed to be
small compared with horizontal dimensions, we use a quasi-two-dimensional (Q2D)
numerical model that only considers the component of the applied magnetic field
normal to the plane of motion. This component can be expressed as
B
z
(
x
,
y
,
z
)
=
B
(
x
,
y
)
g
(
z
),
(1)
where
B
reproduces the variation of the magnetic field in the
x
-
y
plane and is
modeled by a dipolar field distribution created by a square magnetized surface uni-
formly polarized in the normal direction, for which an explicit analytical expression
is available (McCaig
1977
; Cuevas et al.
2006
). In fact, the shape of the magnets is
irrelevant provided the plane of flow is separated from the surface of the magnet, so
that border effects are smoothed out (Figueroa et al.
2009
). In turn,
g
(
x
,
y
)
)
models the decay of the magnetic field in the normal direction
z
(normalized by the
layer thickness
h
), where
(
z
)
=
exp
(
−
ʳ
z
51 is an empirical constant obtained from fitting the
decay of the magnetic field in the vertical direction (Beltrán
2010
) with experimental
data from a permanent magnet (Honji
1991
). In addition, the Q2D model assumes
that the momentum transfer through the thin electrolytic layer is mainly diffusive so
that the velocity field can be expressed as
ʳ
=
0
.
u
(
x
,
y
,
z
,
t
)
=[
u
(
x
,
y
,
t
)
f
(
x
,
y
,
z
),
v
(
x
,
y
,
t
)
f
(
x
,
y
,
z
),
0
]
,
(2)
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