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Why
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Drag
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Analysis
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is
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Necessary
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Observation
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of
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the
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tube
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settlers
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yielded
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interest
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in
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floc
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buildup
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and
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floc
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flow
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in
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tubes.
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As
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the
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flocs
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began
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to
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build
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up,
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some
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started
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to
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roll
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up
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the
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tubes
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and
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flow
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out
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into
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the
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effluent
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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
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\large\[V_\alpha\] |
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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
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Gradients
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The
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Reynolds
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number
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and
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entrance
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region
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length
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were
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calculated
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to
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determine
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whether
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the
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flow
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through
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the
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tubes
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was
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transient
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or
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laminar.
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With
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Reynolds
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numbers
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below
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100
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the
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length
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of
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the
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entrance
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region
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was
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determined
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by
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the
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following
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equation:
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\large
$l_e = 0.06{\mathop{\rm Re}\nolimits} \cdot d$
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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.
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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:
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\large
$\frac{{\partial v}}{{\partial r}} = \frac{{4 \cdot V}}{R}$
{latex}
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This
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equation
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was
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evaluated
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at
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R,
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the
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radius
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of
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the
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tube,
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to
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find
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the
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maximum
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velocity
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gradient
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at
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the
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tube
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walls.
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Table
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1
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lists
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the
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velocity
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gradient
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values
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for
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each
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tube
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at
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the
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test
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critical
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velocities.
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{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
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order
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to
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determine
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a
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minimum
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plate
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spacing
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for
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the
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tanks
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in
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AguaClara
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plants,
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a
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Navier
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Stokes
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equation
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for
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laminar
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flow
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between
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two flat plates
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was
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used.
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{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:
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\large
$\frac{{\partial u}}{{\partial y}} = \frac{{3V}}{{2h}}$
{latex}
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The
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minimum
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spacing
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for
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plate
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settlers
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can
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be
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determined
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using
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the
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above
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equations.
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The
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velocity
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gradients
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found
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in
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each
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tube
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over
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the
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range
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of
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critical
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velocities
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can
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be
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found
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in
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Table
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1.
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By
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comparing
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the
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velocity
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gradients
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in
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table
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1
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and
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the
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results
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table
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from
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the
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...
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it
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can
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be
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determined
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that
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once
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the
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velocity
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gradient
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in
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the
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tube
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reaches
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a
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certain
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value,
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failure
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occurs.
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From
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the
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data
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it
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appears
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that
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failure
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occurs
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around
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2.4
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1/s,
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as
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velocity
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gradients
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beyond
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this
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value
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correspond
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with
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failure
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in
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the
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two
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smallest
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tube
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sizes
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tested.
Drag on a floc
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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
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2
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shows
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the
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force
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balance
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on
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a
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floc.
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As
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stated
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before,
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it
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is
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believe
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that
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when
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the
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drag
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force
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on
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a
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floc
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exceeds
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force
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due
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to
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gravity,
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the
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floc
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beings
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to
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roll
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up
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the
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tube.
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