Environmental Engineering Reference
In-Depth Information
⎧
( )
=
yy
r
⎪
⎪
⎪
1
1
( )
=
yy
r
(5.76)
⎨
2
2
⎪
⎪
⎪
d
dr
y
=
0
at r
=
0
⎩
Where,
y
2
is the surface potential on the channel wall,
y
1
is the potential at
the interface between the two fluids,
r
1
is the radius of inner layer (oil layer),
and
r
2
is the radius of the entire channel, as shown in figure 5.7. The electric
potential distribution in the channel is found by solving equation 5.74 under
boundary conditions specified in equation 5.76 as (Liu et al., 2009):
)
(
⎧
Ikr
Ikr
( )
=
r
01
for
r
r
yy
0
≤≤
⎪
(
)
1
1
(5.77)
011
⎨
(
)
(
)
Ikr
Ikr
I
(
kr r
Ikr
−
2
⎪
⎪
( )
=
02
0
2
1
yy
r
)
+
y
for r
≤≤
r
r
(
(
)
2
1
1
2
⎩
022
021
Figure 5.8 shows the charge distribution in the capillary for the case
when
y
1
=
y
2
= -25 mV and the water saturation is S
w
= 0.8.
The charge density distribution can then be written as:
(
)
⎧
Ikr
Ikr
( )
=−
01
r
r
e
k
2
y
for
0
≤≤
r
r
⎪
(
)
(5.78)
e
o
1
1
011
⎨
(
)
(
)
(
)
Ikr r
Ikr
−
2
Ikr
I
⎪
⎪
( )
=−
02
1
02
r
r
e
k
2
y
+
y
for r
≤≤
rr
(
)
(
)
e
w
2
1
1
2
kr
⎩
0022
021
Where,
I
0
is zero order modified Bessel function of the first kind; and
k
1
and
k
2
are the reciprocal of EDL thickness of the inner and outer layers,
respectively; and
e
0
and
e
w
are the permittivity of the inner and outer layers,
respectively.
Figure 5.9 shows the micro-scale representation of the interfacial forces in
the capillary. At the oil-water interface the continuity of velocity and hydro-
dynamic shear stress can be expressed as (Gao et al., 2007; Liu et al., 2009):
vv
=
=−
( )
∇
⎧
⎨
(5.79)
w
o
⎩
ttr
1
rE
o
we
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