...
Initial
...
Research:
...
In
...
order
...
to
...
help
...
us
...
develop
...
our
...
clear
...
well
...
design,
...
we
...
conducted
...
extensive
...
literature
...
and
...
online
...
research
...
on
...
granular
...
filtration
...
and
...
backwash.
...
We
...
found
...
"Surface
...
Water
...
Treatment
...
for
...
Communities
...
in
...
Developing
...
Countries"
...
by
...
Christopher
...
R.
...
Schulz
...
and
...
Daniel
...
A.
...
Okun
...
and
...
"Physicochemcial
...
Processes
...
for
...
Water
...
Quality
...
Control"
...
by
...
Walter
...
J.
...
Weber,
...
Jr.
...
to
...
be
...
the
...
two
...
most
...
useful
...
sources
...
of
...
information
...
in
...
our
...
research.
...
"Surface
...
Water
...
Treatment
...
for
...
Communities
...
in
...
Developing
...
Countries"
...
From
...
Christopher
...
R.
...
Schulz
...
and
...
Daniel
...
A.
...
Okun,
...
we
...
learned
...
the
...
following
...
useful
...
information
...
with
...
regards
...
to
...
filtration:
...
•
...
Filtration
...
is
...
the
...
separation
...
of
...
suspended
...
impurities
...
from
...
water
...
by
...
passage
...
through
...
porous
...
media.
...
•
...
Slow
...
sand
...
filtration
...
consists
...
of
...
slowly
...
filtering
...
water
...
through
...
a
...
layer
...
of
...
ungraded
...
fine
...
sand.
...
Periodically,
...
the
...
top
...
layer
...
is
...
clogged
...
by
...
impurities
...
and
...
is
...
skimmed
...
off
...
the
...
top.
...
•
...
Rapid
...
sand
...
filtration
...
rapidly
...
conducts
...
filtration
...
in
...
depth
...
as
...
compared
...
to
...
the
...
slow
...
sand
...
filter
...
which
...
uses
...
only
...
the
...
top
...
layer
...
to
...
capture
...
suspended
...
particles.
...
A
...
lighter
...
anthracite
...
coal
...
layer
...
with
...
larger
...
pore
...
spaces
...
than
...
sand
...
is
...
used
...
on
...
top
...
of
...
a
...
sand
...
layer
...
to
...
capture
...
larger
...
particles
...
while
...
allowing
...
the
...
smaller
...
particles
...
passage
...
to
...
be
...
captured
...
by
...
the
...
lower
...
sand
...
layer.
...
•
...
Backwashing
...
is
...
the
...
act
...
of
...
removing
...
the
...
captured
...
impurities
...
in
...
the
...
filter
...
bed
...
by
...
introducing
...
enough
...
water,
...
usually
...
from
...
the
...
effluent
...
end,
...
to
...
fluidize
...
and
...
expand
...
the
...
bed
...
and
...
wash
...
away
...
the
...
now
...
released
...
impurities.
...
Backwashing
...
is
...
an
...
art.
...
There
...
is
...
neither
...
a
...
set
...
time
...
nor
...
a
...
set
...
amount
...
of
...
backwash
...
water
...
required.
...
If
...
a
...
filter
...
is
...
heavily
...
clogged,
...
significant
...
length
...
of
...
backwash
...
and
...
greater
...
backwash
...
water
...
for
...
greater
...
bed
...
expansion
...
are
...
required.
...
If
...
the
...
influent
...
water
...
is
...
relatively
...
low
...
in
...
NTU,
...
less
...
clogging
...
may
...
occur
...
and
...
a
...
shorter
...
backwash
...
and
...
less
...
water
...
may
...
be
...
required.
...
Consequently,
...
rapid
...
filtration
...
and
...
required
...
backwash
...
operations
...
necessitate
...
a
...
well
...
trained
...
crew
...
of
...
operators.
...
This
...
finding
...
leads
...
us
...
to
...
define
...
our
...
objective
...
as
...
not
...
only
...
providing
...
a
...
clear
...
well
...
design
...
that
...
works
...
but
...
a
...
set
...
of
...
instructions
...
in
...
operating
...
that
...
design.
...
•
...
Headloss
...
via
...
expanded
...
media
...
can
...
be
...
calculated
...
as
...
below:
| Latex |
|---|
} \large $$ h = D(1 - f)(p - 1) $$ {latex} |
where:
...
h
...
=
...
headloss
...
across
...
the
...
fluidized
...
bed
...
(m)
...
D
...
=
...
unexpanded
...
bed
...
depth
...
(m)
...
f
...
=
...
porosity
...
of
...
unexpanded
...
bed
...
(dimensionless)
...
p
...
=
...
specific
...
gravity
...
of
...
the
...
filter
...
medium
...
(dimensionless)
...
Once
...
fluidized,
...
headloss
...
through
...
an
...
expanded
...
bed
...
is
...
constant.
...
"Physicochemcial
...
Processes
...
for
...
Water
...
Quality
...
Control"
...
From
...
Walter
...
J.
...
Weber,
...
Jr.,
...
we
...
learned
...
the
...
following
...
useful
...
information
...
with
...
regards
...
to
...
filtration.
...
These
...
are
...
empiricial
...
equations
...
for
...
calculating
...
the
...
minimum
...
velocity
...
to
...
fluidize
...
a
...
filter
...
bed
...
for
...
backwash
...
and
...
the
...
velocity
...
required
...
for
...
a
...
specific
...
degree
...
of
...
bed
...
expansion.
...
•
...
Minimal
...
Fluidization
...
Velocity
...
Equation:
| Latex |
|---|
}
\large
$$
V_f = {{0.00381(d_{60} ){}^{1.82}\{ \omega _s (\omega _m - \omega _s )\} ^{0.94} } \over {\mu ^{0.88} }}
$$
|
where
| Latex |
|---|
{latex} where {latex}\large$$V_f $${latex} |
=
...
fluidization
...
velocity,
...
gpm/square
...
feet
| Latex |
|---|
}\large$$\omega _s $${latex} |
=
...
specific
...
weight
...
of
...
water,
...
lb/cubic
...
feet
| Latex |
|---|
}\large$$\omega _s $${latex} |
=
...
specific
...
weight
...
of
...
water,
...
lb/cubic
...
feet
| Latex |
|---|
}\large$$d_{60}$${latex} |
=
...
diameter
...
of
...
which
...
60%
...
of
...
the
...
media
...
is
...
equal
...
to
...
or
...
smaller,
...
mm
| Latex |
|---|
}\large$$\mu $${latex} |
=
...
viscosity
...
of
...
water,
...
centipose
...
•
...
Bed
...
Expansion
...
Equation:
| Latex |
|---|
} \large $$ \overline \varepsilon = 1 - {D \over {D_e }}(1 - \varepsilon ) $$ {latex} where |
where
| Latex |
|---|
{latex}\large$$\overline \varepsilon$${latex} |
=
...
porosity
...
of
...
expanded
...
bed
| Latex |
|---|
}\large$$\varepsilon $${latex} |
=
...
porosity
...
of
...
unexpanded
...
bed
...
D
...
=
...
depth
...
of
...
unexpanded
...
bed
| Latex |
|---|
}\large$$D_e$${latex} |
=
...
depth
...
of
...
expanded
...
bed
...
•
...
Fluidization
...
Velocity
...
Equation:
| Latex |
|---|
} \large $$ V = K_e (\overline \varepsilon )^{n_e } $$ {latex} where V = fluidization velocity |
where
V = fluidization velocity
| Latex |
|---|
{latex}\large$$K_e,n_e$${latex} |
=
...
Constants
...
derived
...
experimentally
...
or
...
empirically
...
as
...
shown
...
below
| Latex |
|---|
\\ {latex}\large$$ n_e = 4.45{\mathop{\rm Re}\nolimits} _0^{ - 0.1} $$ {latex} where {latex} |
where
| Latex |
|---|
\large$$
{\mathop{\rm Re}\nolimits} _0 = {{\rho _l \cdot 8.45 \cdot V_f \cdot d_{60} } \over \mu }
$$
{latex}
|
where:
...
μ
...
= the
...
dynamic
...
viscosity
...
of
...
the
...
fluid
...
(Pa
...
-
...
s
...
or
...
N
...
-
...
s/m²
...
or
...
kg/m
...
-
...
s)
...
p=
...
density
...
of
...
the
...
fluid
...
(kg/m^3)
d60= the particle diameter at which 60% of the particles are smaller or equal to.
Vf= minimum fluidization velocity of the media
and
| Latex |
|---|
and {latex}\large$$ K_e = {{V_f } \over {\varepsilon ^{n_e } }} $$ {latex} |