Methods
Our design process consisted of 3 major steps. First we designed the individual filter bed itself based on the relationship equations mentioned in the theory section. Second, we sized the pipe in our system so that the head loss experienced by the pipes is never greater than 10% of the head loss experienced by the sand. If the head loss in the sand is not greater than the head loss in the pipes, there will be preferential flow and not all of the pipes in the manifold will have equal flow. Third step was to design the minimum distance between the entrance pipes from the sedimentation tank to the height of the gutter to ensure proper back wash.
In our first step, we chose a conservative back wash velocity 14 mm/s that we know empirically to raise a layer of sand bed 30%. We also chose a conservative back wash flow rate of 3.15 L/s which is half of the maximum plant flow rate at Agalteca. Based on that given parameter, we solved for the area of the filtration bed as shown below.
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h3. Methods Our design process consisted of 3 major steps. First we designed the individual filter bed itself based on the relationship equations mentioned in the theory section. Second, we sized the pipe in our system so that the head loss experienced by the pipes is never greater than 10% of the head loss experienced by the sand. If the head loss in the sand is not greater than the head loss in the pipes, there will be preferential flow and not all of the pipes in the manifold will have equal flow. Third step was to design the minimum distance between the entrance pipes from the sedimentation tank to the height of the gutter to ensure proper back wash. In our first step, we chose a conservative back wash velocity 14 mm/s that we know empirically to raise a layer of sand bed 30%. We also chose a conservative back wash flow rate of 3.15 L/s which is half of the maximum plant flow rate at Agalteca. Based on that given parameter, we solved for the area of the filtration bed as shown below. {latex}$A_{Filter} = {{Q_{BW} } \over {V_{BW} }} = {{3.15L/s} \over {14mm/s}} = .225m^2 ${latex} |
We
...
made
...
the
...
design
...
a
...
square
...
with
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a
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side
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length
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of
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0.474m
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which
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would
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also
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be
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the
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length
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of
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the
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filtration
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inlet
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and
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outlet
...
tubes.
...
In
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our
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second
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step,
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we
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solved
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for
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the
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head
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loss
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that
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occurs
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in
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the
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20
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layers
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of
...
sand
...
using
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the
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Carmen-Kozeny
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Equation
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shown
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below:
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}$$ {{h_l } \over L} = 36k{{\left( {1 - \varepsilon } \right)^2 } \over {\varepsilon ^3 }}{{\nu V_a } \over {gd_{sand}^2 }} $${latex} Where |
Where
h=head
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loss.
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L=length
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of
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layer
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which
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in
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this
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case
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is
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20
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cm.
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K=Kozeny
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constant
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of
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5.
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}$$ \varepsilon $$ {latex} |
=porosity
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of
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sand=0.4
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for
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our
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case.
...
g=acceleration
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of
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gravity.
...
d=diameter
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of
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sand
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which
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in
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this
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case
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is
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0.55mm.
...
V=velocity
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of
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the
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fluid
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passing
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through
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the
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filter
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ν=kinematic
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viscosity=10-6
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m2/s
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We
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calculated
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that
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the
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head
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loss
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is
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around
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10cm.
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Since
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we
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want
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the
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head
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loss
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in
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the
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sand
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to
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dominate,
...
we
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design
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all
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of
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our
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pipes
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to
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have
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large
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enough
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diameter
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so
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that
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their
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head
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loss
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never
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exceeds
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10%
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of
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the
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sand's
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head
...
loss.
...
We
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utilized
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the
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manifold
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equation
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as
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shown
...
below:
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}$$
\sum\limits_{i = 1}^{n - 1} {h_{{\rm{f}}_i } } = {\rm{f}}_i {{L_M } \over {D_M }}{1 \over {2g}}\left( {{{Q_M } \over {A_M }}} \right)^2 {1 \over {\left( {n - 1} \right)n^2 }}\sum\limits_{i = 1}^{n - 1} {\left( {n - i} \right)^2 }
$$ |
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{latex} {latex} $$L_M = L_P \left( {n - 1} \right)$${latex} |
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{latex} $$ \sum\limits_{i = 1}^{n - 1} {\left( {n - i} \right)^2 } = {{n\left( {n - 1} \right)\left( {2n - 1} \right)} \over 6} $${latex} |
Where
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the
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variables
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not
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defined
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previously
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are
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as
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follows:
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N
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=
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number
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of
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ports
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M
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=
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Manifold
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P
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=
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Port
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D
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=
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Diameter
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as
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dictated
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by
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the
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subscript
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F
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=friction
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factor
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As
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the
...
...
...
shows,
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we
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calculated
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the
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inlet
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and
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outlet
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tubes
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to
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be
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½
...
inch
...
and
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the
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manifold
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connecting
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them
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to
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be
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3
...
inches
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in
...
diameter.
...
Because
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the
...
entire
...
back
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wash
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flow
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rate
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must
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pass
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through
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the
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inlet
...
tubes
...
during
...
back
...
wash,
...
the
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inlet
...
tubes
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of
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the
...
bottom
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plane
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need
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to
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be
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0.75
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inch
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in
...
diameter.
...
The
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bottom
...
manifold
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and
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the
...
rest
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of
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the
...
pipe
...
system
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need
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to
...
be
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3
...
inch
...
in
...
diameter.
...
Our
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last
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step
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was
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to
...
determine
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the
...
minimum
...
distance
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between
...
the
...
entrance
...
pipes
...
from
...
the
...
sedimentation
...
tank
...
to
...
the
...
gutter
...
of
...
the
...
filtration
...
unit.
...
The
...
minimum
...
distance
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must
...
be
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greater
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than
...
the
...
head
...
loss
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that
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occurs
...
through
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the
...
pipes
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and
...
the
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expanded
...
bed
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during
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head
...
loss
...
as
...
shown
...
below
...
in
...
Figure
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4
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Distance
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from
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Entrance
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Tank
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to
...
Gutter
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and
...
in
...
our
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MathCAD
...
file.
...
We
...
first
...
calculated
...
the
...
head
...
loss
...
in
...
the
...
pipes
...
during
...
back
...
wash
...
to
...
be
...
6.7
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cm.
...
Most
...
of
...
the
...
head
...
loss
...
occurs
...
in
...
the
...
expanded
...
bed
...
calculated
...
as
...
shown
...
below,
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an
...
Okun
...
equation,
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as
...
shown
...
on
...
the
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Review
...
of
...
Existing
...
Research
...
section:
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|---|
} $h = H(1 - \varepsilon )(SG - 1)$ {latex} |
Where
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the
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previously
...
undefined
...
variables
...
are
...
h=
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Head
...
Loss
...
H=Height
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of
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Unexpanded
...
Bed
...
SG=specific
...
gravity
Figure 1: Minimum Distance from Gutter to Entrance Pipe
We calculate the head loss to be 2.145 m which means that there must be at least 1.1 m distance from the entrance pipe to the gutter. The effluent pipe of the filtration tank should be slightly higher than the top of the sand filter to ensure that even during the no flow the sand would stay wet. The high elevation of the effluent pipe would essentially trap the water in the filter until the flow of water pushes it over during normal filtration operations.