...
Why
...
Drag
...
Analysis is Necessary
Observation of the tube settlers yielded interest in floc buildup and floc flow in tubes. As the flocs began to build up, some started to roll up the tubes and flow out into the effluent instead of falling back into the floc blanket. It was determined that a drag force was possibly preventing the flocs from settling out. The question remained, however, as to why smaller tubes at higher flow rates experienced the rolling flocs but not the larger tubes.
It was determined that velocity gradients vary with the tube diameter, and that the drag force was related to the velocity gradient at the tube wall. As the diameter decreased at the same
| Latex |
|---|
\large\[V_\alpha\] |
| Wiki Markup |
|---|
is necessary Observation of the tube settlers yielded interest in floc buildup and floc flow in tubes. As the flocs began to build up, some started to roll up the tubes and flow out into the effluent instead of settling out and fall back into the floc blanket. It was determined that a drag force was possibly preventing the flocs from settling out. The question remained, however, as to why smaller tubes at higher flow rates experienced the rolling flocs but not the larger tubes {float:right|border=2px solid black} !Plate Settler Spacing Research Fall 2008^Uniformflow.png|width=200px|height=250px! *h5. Figure 1: Development of Uniform Flow*fully developed flow {float} h3. |
Velocity
...
Gradients
...
The
...
Reynolds
...
number
...
and
...
entrance
...
region
...
length
...
were
...
calculated
...
to
...
determine
...
whether
...
the
...
flow
...
through
...
the
...
tubes
...
was
...
transient
...
or
...
laminar.
...
With
...
Reynolds
...
numbers
...
below
...
100
...
the
...
length
...
of
...
the
...
entrance
...
region
...
was
...
determined
...
by
...
the
...
following
...
equation:
| Latex |
|---|
\large
$l_e = 0.06{\mathop{\rm Re}\nolimits} \cdot d$
|
It was then determined that the flow though the tubes became fully developed very quickly.
Figure 1 shows the evolution from uniform to fully developed. The parabola represents the fully developed velocity profile through the tubes. It is clear from this image that flocs experience higher velocity gradients in the entrance region.
The next calculation involved the Navier Stokes equation for laminar flow through a cylindrical tube, as seen below.
| Latex |
|---|
INSERT EQUATION It was then determined that the flow though the tubes became laminar very quickly. Figure 1 shows the evolution flow from uniform to laminar. The parabola represents the velocity profile through the tubes. It is clear from this image that flocs experience high velocity gradients along the sides of the profile. The next calculation involved the Navier Stokes equation for laminar flow through a cylindrical tube, as seen below. {latex} \large $\frac{{\partial v}}{{\partial r}} = \frac{1}{\mu }\left( {\frac{{\partial p}}{{\partial z}}} \right)R + c_1 $width=250px {latex} Final equation: {latex}$ |
Final equation:
| Latex |
|---|
\large
$\frac{{\partial v}}{{\partial r}} = \frac{{4 \cdot V}}{R}$
{latex}
|
This
...
equation
...
was
...
evaluated
...
at
...
R,
...
the
...
radius
...
of
...
the
...
tube,
...
to
...
find
...
the
...
maximum
...
velocity
...
gradient
...
at
...
the
...
tube
...
walls.
...
Table
...
1
...
lists
...
the
...
velocity
...
gradient
...
values
...
for
...
each
...
tube
...
at
...
the
...
test
...
critical
...
velocities.
| Wiki Markup |
|---|
{float:right|border=2px solid black|width=700} {excel:file=PSS flow rate experiment^DataAnalysis_aguaclara.xls |sheet=Velocity Gradient Table} *Table 1. Results: The Velocity Gradients for the Tested Critical Velocities* {float} |
In
...
order
...
to
...
determine
...
a
...
minimum
...
plate
...
spacing
...
for
...
the
...
tanks
...
in
...
AguaClara
...
plants,
...
a
...
Navier
...
Stokes
...
equation
...
for
...
laminar
...
flow
...
between
...
two flat plates
...
was
...
used.
| Latex |
|---|
{latex} \large $\frac{{\partial u}}{{\partial y}} = \frac{1}{\mu }\left( {\frac{{\partial p}}{{\partial x}}} \right)y + c_1 $width=250px {latex} Final equation: {latex}$ |
Final equation:
| Latex |
|---|
\large
$\frac{{\partial u}}{{\partial y}} = \frac{{3V}}{{2h}}$
{latex}
|
The
...
minimum
...
spacing
...
for
...
plate
...
settlers can be determined using the above equations.
The velocity gradients found in each tube over the range of critical velocities can be found in Table 1. By comparing the velocity gradients in table 1 and the results table from the flow rate experiment it can be determined that once the velocity gradient in the tube reaches a certain value, failure occurs. From the data it appears that failure occurs around 2.4 1/s, as velocity gradients beyond this value correspond with failure in the two smallest tube sizes tested.
Drag on a floc
| Wiki Markup |
|---|
was determined to be TBD (FIND USING MATHCAD) The velocity gradients found in each tube over the range of critical velocities can be found in Table 1. By comparing the velocity gradients in table 1 and the results table from the [flow rate experiment|PSS flow rate experiment] it can be determined that once the velocity gradient in the tube reaches a certain value, failure occurs. From the data it appears that failure occurs around 2.4 1/s, as velocity gradients beyond this value correspond with failure in the two smallest tube sizes tested. h3. Drag on a floc {float:right|border=2px solid black} !Drag on floc.png|width=90px|height=90px! *Figure 2: Force Balance on a floc* {float} |
Figure
...
2
...
shows
...
the
...
force
...
balance
...
on
...
a
...
floc.
...
As
...
stated
...
before,
...
it
...
is
...
believe
...
that
...
when
...
the
...
drag
...
force
...
on
...
a
...
floc
...
exceeds
...
force
...
due
...
to
...
gravity,
...
the
...
floc
...
beings
...
to
...
roll
...
up
...
the
...
tube.
...