...
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 |
---|
ft^2 {latex}\large$$\omega _s $${latex} |
=
...
specific
...
weight
...
of
...
water,
...
lb/cubic feet
Latex |
---|
ft^3 {latex}\large$$\omega _s $${latex} |
=
...
specific
...
weight
...
of
...
water,
...
lb/cubic feet
Latex |
---|
ft^3 {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
$$
\overline \varepsilon = 1 - {D \over {D_e }}(1 - \varepsilon )
$$
|
where
Latex |
---|
\large$$\overline \varepsilon$$ |
= porosity of expanded bed
Latex |
---|
\large$$\varepsilon $$ |
= porosity of unexpanded bed
D = depth of unexpanded bed
Latex |
---|
\large$$D_e$$ |
= depth of expanded bed
• Fluidization Velocity Equation:
Latex |
---|
\large
$$
V = K_e (\overline \varepsilon )^{n_e }
$$
|
where
V = fluidization velocity
Latex |
---|
\large$$K_e,n_e$$ |
= Constants derived experimentally or empirically as shown below
Latex |
---|
\large$$
n_e = 4.45{\mathop{\rm Re}\nolimits} _0^{ - 0.1}
$$
|
where
Latex |
---|
\large$$
{\mathop{\rm Re}\nolimits} _0 = {{\rho _l \cdot 8.45 \cdot V_f \cdot d_{60} } \over \mu }
$$
|
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 |
---|
\large$$ K_e = {{V_f } \over {\varepsilon ^{n_e } }} $$ centipose • INPUT MATHTYPE Equation 4-50 to 4-53. Fluidization Velocity to achieve different degrees of expansion.(Weber 173) |