...
Plate
...
Settler
...
Spacing
...
Figure
...
1
...
shows
...
the
...
floc
...
size
...
that
...
could
...
be
...
captured
...
by
...
different
...
tube
...
settler
...
diameters.
...
When
...
the
...
ratio
...
has
...
a
...
value
...
of
...
1,
...
the
...
sedimentation
...
velocity
...
matches
...
the
...
upflow
...
velocity
...
of
...
the
...
floc
...
particle,
...
and
...
this
...
is
...
defined
...
as
...
the
...
critical
...
diameter,
...
i.e.
...
the
...
floc
...
diameter
...
in
...
which
...
floc
...
roll
...
up
...
will
...
begin,
...
for
...
each
...
of
...
the
...
given
...
tube
...
diameters
...
and
...
plate
...
settler
...
spacing.
Figure 1: The ratio of Sedimentation Velocity to Fluid Velocity vs. Floc Diameter
The graph is cut off at a particle size of 1 mm that is on the order of magnitude of size of a colloidal particle. For plate spacings greater than or equal to 2 cm and for tube diameters greater than or equal to 5 cm, it is predicted that there will be no floc roll up under AguaClara conditions.
Calculation of the Minimum Diameter of the Flocs that Settle from the Sedimentation Velocity Equation
Assuming an upward flow velocity of 1.2 mm/s, which used in the newer AguaClara plants, the diameter of floc that will roll-up was determined by using a root finding algorithm, and the plate settler spacing or tube diameter was plotted versus the minimum floc diameter. The minimum floc diameter corresponds to the minimum size of a floc particle that will roll up into the effluent; or the maximum size of a floc particle that the plate settler will prevent from going into the effluent.
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!Plate spacing vs floc diameter.png\|width=700px,align=centre! |
...
Figure
...
2:
...
Plate
...
Spacing
...
or
...
Tube
...
Diameter
...
vs.
...
Minimum
...
Floc Diameter
The minimum floc diameter corresponds to the minimum size of particles that will still settle out of the tube and return to the floc blanket instead of going into the effluent. With larger plate settler spacing, most floc roll-up could theroetically be eliminated.
The minimum floc diameter that will be captured for a given upflow velocity and tube settler diameter can be calculated in the equation below. This equation uses a simplification that the velocity profile is linear, not parabolic, near the wall. This linearized approach produces very similiar results (Perhaps statistically quantify this with some sample diameters.
Figure 3 graphically displays the linear velocity gradient solutions. The curves are not quite straight on a log log plot. This is due to the quadratic in the velocity profile. We can obtain a very good approximation by using the velocity gradient at the wall and assuming a linear velocity gradient. That assumption makes an analytical solution possible.
Figure 3: Floc Diameter vs. Spacing
For small floc sizes this linearization is valid and produces an analytical solution. For larger flocs that could roll-up, the linearization is invalid because the slope tends more and more parabolic closer the the center. However, Figure 4 illustrates that linearized equations show that with smaller tubes, the size of a floc particle that will roll up into the effluent varies very little with vertical velocity.
Figure 4: Floc Spacing vs. Floc Diameter
| Wiki Markup |
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Diameter_ \\ \\ The minimum floc diameter corresponds to the minimum size of particles that will still settle out of the tube and return to the floc blanket instead of going into the effluent. With larger plate settler spacing, most floc roll-up could theroetically be eliminated. The minimum floc diameter that will be captured for a given upflow velocity and tube settler diameter can be calculated in the equation below. This equation uses a simplification that the velocity profile is linear, not parabolic, near the wall. This linearized approach produces very similiar results (Perhaps statistically quantify this with some sample diameters. Figure 3 graphically displays the linear velocity gradient solutions. The curves are not quite straight on a log log plot. This is due to the quadratic in the velocity profile. We can obtain a very good approximation by using the velocity gradient at the wall and assuming a linear velocity gradient. That assumption makes an analytical solution possible. !Floc diameter v. Spacing.png|width=700px,align=centre! _Figure 3: Floc Diameter vs. Spacing_ For small floc sizes this linearization is valid and produces an analytical solution. For larger flocs that could roll-up, the linearization is invalid because the slope tends more and more parabolic closer the the center. However, Figure 4 illustrates that linearized equations show that with smaller tubes, the size of a floc particle that will roll up into the effluent varies very little with vertical velocity. !Plate spacing v. floc diameter.png|width=700px,align=centre! _Figure 4: Floc Spacing vs. Floc Diameter_ {latex} \large $$ d = d_0 \left( {{{18V_t \Phi \nu _{H_2 O} } \over {gd_0^2 }}{{\rho _{H_2 O} } \over {\rho _{Floc_0 } - \rho _{H_2 O} }}} \right)^{{1 \over {D_{Fractal} - 1}}} $$ {latex} \\ |
The
...
critical
...
velocity
...
model
...
can
...
be
...
utilized
...
to
...
calculate
...
the
...
desired
...
spacing
...
to
...
capture
...
a
...
floc
...
particle
...
of
...
a
...
particular
...
size.
...
The
...
following
...
equation
...
results
...
were
...
summarized
...
in
...
Figure
...
5.
...
Figure
...
5
...
represents
...
the
...
minimum
...
spacing
...
that
...
will
...
capture
...
a
...
floc
...
particle
...
with
...
a
...
particular
...
settling
...
velocity.
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{latex} \large $$ S = V_{up} {{108\Phi \nu _{H_2 O} d^2 } \over {g\sin ^2 (\alpha )d_0^3 }}\left( {{{d_0 } \over d}} \right)^{D_{Fractal} } {{\rho _{H_2 O} } \over {\rho _{Floc_0 } - \rho _{H_2 O} }} $$ {latex} |
Where:
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|---|
{latex}\large$$ alpha $${latex} |
=
...
The
...
angle
...
of
...
the
...
tube
...
settler
...
(60
...
degrees)
Figure 5: Minimum spacing vs. Floc Sedimentation Velocity
Figure 6 represents the absolute minimum plate settler spacing that will capture floc particles with a settling velocity of 0.12 mm/s in an AguaClara plant. Theoretically, any spacing below the intersection of two lines would produce a worsened effluent turbidity.
Figure 6: Minimum Plate Settler Spacing vs. Capture Velocity