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Calculation
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of
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Ratio
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of
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Settling
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Velocity
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to
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Particle
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Velocity
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The
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calculation
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of
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this
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ratio
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is
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very
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important
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in
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order
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to
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begin
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modeling
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floc
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roll
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up.
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This
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ratio
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is
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the
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key
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to
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determining
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whether
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or
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not
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floc
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roll
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up
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will
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occur
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for
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a
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given
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set
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of
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conditions.
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In
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order
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to
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determine
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the
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critical
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velocity
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at
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which
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floc
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particles
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will
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begin
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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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and
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into
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the
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effluent,
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we
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compare
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the
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settling
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velocity
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with
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the
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particle
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velocity
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experienced
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from
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the
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velocity
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gradient.
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The
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settling
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velocity
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of
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a
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particle
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in
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a
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tube
...
settler
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can
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be
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expressed
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as
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follows
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(Munson,
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1998)
| Wiki Markup |
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{latex} \large $$ V_t = {{gd_0 ^{\left( {3 - D_{Fractal} } \right)} d^{\left( {D_{Fractal} - 1} \right)} } \over {18\Phi \nu }}\left( {{{\rho _{Floc_0 } } \over {\rho _{H_2 O} }} - 1} \right) $$ {latex} |
Where:
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g
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= Gravity
| Wiki Markup |
|---|
Gravity {latex}\large $$d_o $$ {latex} |
=
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size
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of
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the
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primary particles
| Wiki Markup |
|---|
particles {latex}\large $$D_{fractal} $$ {latex} |
=
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fractal
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dimension
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of
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the
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floc particles
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|---|
particles {latex}\large $$\Phi $$ {latex} |
=
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shape
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factor
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for
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drag
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on
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flocs
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which
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is
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equal to
| Wiki Markup |
|---|
to {latex}\large $$\nu $$ {latex} |
= viscosity
| Wiki Markup |
|---|
viscosity {latex}\large $$\rho _{floc}$$ {latex} |
=
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density
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of
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the
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floc particle
| Wiki Markup |
|---|
particle {latex}\large $$\rho _{H_2 O} $$ {latex} |
=
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density
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of
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water
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The
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particle
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velocity
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experienced
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as
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a
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result
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of
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the
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velocity
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gradient
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can
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be
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expressed
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as
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follows
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(Munson,
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1998)
| Wiki Markup |
|---|
{latex} \large $$ V_{particle} = V_{ratio} V_\alpha \left[ {1 - \left( {{{{{d_{tube} } \over 2} - d_{Floc} } \over {{{d_{Tube} } \over 2}}}} \right)^2 } \right] $$ {latex} Where |
Where
| Wiki Markup |
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{latex}\large $$V_\alpha $$ {latex}
|
=
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directional
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velocity
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in
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the
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tube settler
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settler {latex}\large $$d_{tube}$$ {latex} |
=
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diameter
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of
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the
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tube settler
| Wiki Markup |
|---|
settler {latex}\large $$ d_{floc} $$ {latex} |
=
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the
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diameter
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of
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floc
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particles
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Vratio
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=
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the
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maximum
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velocity
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at
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the
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center
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of
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the
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tube-
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for
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a
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plate
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settler
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this
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value
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is
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1.5
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times
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the
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average
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velocity.
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For
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a
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tube
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settler
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this
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value
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is
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2.
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Therefore,
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the
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ratio
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between
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the
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settling
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velocity
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of
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the
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particle
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and
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the
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velocity
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experienced
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as
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a
...
result
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of
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the
...
velocity
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gradient
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can
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be
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expressed
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as
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by
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the
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below
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equation.
| Wiki Markup |
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{latex} \large $$ \Pi _V = {{{{g\sin (\alpha )d_0 ^2 } \over {18\Phi \nu }}{{\rho _{Floc_0 } - \rho _{H_2 O} } \over {\rho _{H_2 O} }}\left( {{{d_{Floc} } \over {d_0 }}} \right)^{D_{Fractal} - 1} } \over {V_{ratio} {{V_{Up} } \over {\sin (\alpha )}}\left[ {1 - \left( {{{{{d_{Tube} } \over 2} - d_{Floc} } \over {{{d_{Tube} } \over 2}}}} \right)^2 } \right]}} $$ {latex} |
This
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ratio
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is
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a
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function
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of
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particle
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diameter,
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tube
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diameter,
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upflow
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velocity
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and
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the
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angle
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of
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the
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plate
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settler.
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When
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this
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ratio
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is
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greater
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than
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one
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(ie
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the
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settling
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velocity
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is
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greater
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than
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the
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velocity
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experienced
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by
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the
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floc
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particles
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in
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the
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tube),
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the
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flocs
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will
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fall
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back
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into
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floc
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blanket.
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When
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this
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ratio
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is
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equal
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to
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one,
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the
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particles
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will
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remain
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stationary
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in
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the
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tube
...
settler.
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When
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the
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ratio
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is
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less
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than
...
one,
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the
...
velocity
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of
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the
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particles
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will
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exceed
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the
...
settling
...
velocity
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and
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the
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floc
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particles
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will
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roll
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up
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into
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the
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effluent,
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creating
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a
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highter
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turbidity.
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References
Munson,
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B.,Young,
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D.,
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Okiishi,
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T.,
...
(1998).
...
Fundamentals
...
of
...
Fluid
...
Mechanics
...
(3rd
...
ed.).
...
New
...
York,
...
NY:
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John
...
Wiley
...
&
...
Sons,
...
Inc.
...