Pumping Station Desing - Second Edition by Robert L. Sanks, George Tchobahoglous, Garr M. Jones
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•
Program
the
Moody diagram
on a
computer
by
assuming
it to
consist
of a
family
of
short, straight
lines
[9].
•
Guess
the
pipe size
and
thus estimate
v,
calculate
R,
find /
from
the
Moody diagram,
and
compute
h
from
the
Darcy-Weisbach
formula. Revise
v and D,
if
necessary,
and
recompute.
Other
Pipe Formulas
There
are
many formulas
for flow in
pipes,
but
none
is
easier
to use
than
the
Hazen-
Williams,
and
none
is
more accurate
or
universally applicable than
the
Darcy-Weisbach
supplemented
by the
Moody
dia-
gram
or the
Colebrook equation.
The
limitation
of the
accuracy
of all
pipe formulas lies
in the
estimation
of
the
proper
coefficient
of
friction,
a
value that cannot
be
physically measured
and,
hence,
is
subject
to
large
error
(see
Subsection
"Friction
Coefficients").
Comparison
of
f
and C
The
Darcy-Weisbach
friction
factor
can be
compared
to the
Hazen-Williams
C
factor
by
solving both equa-
tions
for the
slope
of the
hydraulic gradeline
and
equating
the two
slopes. Rearranging
the
terms gives,
in
SI
units,
,
( 1 Y 134
"1
/o
n
x
/ =
[
c
,4
v
o,
V
,
67
J
(3-1*»
where
v is in
meters
per
second
and D is in
meters.
In
U.S. customary
units,
the
relationship
is
/ -
(;.«)(/
<
v'")
<3
-
i4b)
where
v is in
feet
per
second
and D is in
feet.
For any
given
pipe
and
velocity,
the
relation between
^/D
and C
can
be
found
by
calculating
/
from
Equation
3-14
and
entering
the
Moody diagram
with
R
and/to
find
£/£>.
Friction Coefficients
The
major weakness
of any
headloss formula
is the
uncertainty
in
selecting
the
correct
friction
coefficient.
The
proper
friction
factor
for new
pipe
is
uncertain
because
of the
variation
in
roughness
of the
pipe
walls,
quality
of
installation,
effect
of
slight angular
offsets
in
laying
the
pipe,
and
water
quality.
For
exam-
ple,
the H-W C
factor
should
be
reduced
by 5
units
for
pipe laid
in
hilly regions
due to the
angular deflec-
tion
at
joints.
Anticipating
the
friction
factor
after
years
of
ser-
vice
is
doubly
difficult
due to
changes caused
by
cor-
rosion, deposition
of
minerals
or
grease,
or
attachment
of
bacterial slimes. Estimation
of the
friction
coeffi-
cient merits careful attention.
Half
a
century
ago,
unlined cast-iron water
pipe
(or
cast iron with
the
then-common
but
short-lived bitu-
minous
linings)
did
indeed become coated with tuber-
cles
and,
thus, became very rough
in a few
years.
But
the
modern
use of
cement mortar
or
plastic linings
for
ductile iron
and
steel pipe
has
eliminated
the
devastat-
ing
effect
of
rust
and
tuberculation
on
friction. Plastic
pipe remains very smooth unless foreign matter
col-
lects
on the
walls. Water treatment
and
bacterial slime
in
water pipes
and
grease
in
wastewater pipes
can
greatly
increase
the
interior pipe roughness indepen-
dently
of the
pipe material. This subject
is
addressed
in
more detail below.
Hazen-Williams
C
The
basis
of the
Hazen-Williams
C
factor
in
Equation
3-8 has
resulted
in
some
confusion.
The
factor
is a
function
not
only
of the
smoothness
of the
pipe wall,
but
also
of the
difference
between
the
actual
ID of the
pipe
and the
nominal pipe size.
The
calculation
of C
from
field
data
is, by
custom, based
on the
nominal
diameter
of the
pipe.
One
could scarcely
do
otherwise,
because
finding the ID of a
buried pipe
is so
difficult
(see Section
3-9).
Even
if it is
physically measured
at
one
point, there
is no
guarantee
of
uniformity.
The use
of
nominal diameters, however, leads
to
strange
con-
clusions.
For
example,
the C
value
of an
uncoated,
new
Class
50
ductile iron pipe
300 mm (12
in.)
in
diameter
is
typically given
as
130.
A
Class
56
pipe
carries
94% as
much water,
so its C
value should
be
122,
but
such
a
listing
is
unlikely
to be
found.
The
ratio
of
actual
to
nominal diameter accounts
for the
difference
between
the
published
C
values
for
ductile
iron
pipe
(DIP)
and
steel pipe
with
its
smaller bore
when
both
are
lined with cement mortar.
The
confu-
sion
over
the
proper
C
value
to use is
worsened
because
the
nominal diameter
of
steel pipe
300 mm
(12
in.)
and
smaller
is the ID,
whereas
for
larger pipe,
the
nominal diameter
is the OD. To
permit reasonably
accurate
estimation
of
friction
losses,
the C
value
ought
to be
selected
for the
type
of
lining thus allow-
ing
the
true
ID to be
applied
to the
Hazen-Williams
equation.
Matters
are not
improved
by the
apparent increase
of
C
with diameter. According
to
AWWA
Manual
Mil
[10],
the
average value
of C for
pipe with smooth inte-
rior
linings
can be
approximated
as C = 140 +
0.17J,
where
d is
inside pipe diameter
in
inches.
After
a
long
term
of
lining deterioration, slime buildup, etc.,
C =
130 +
0.
16
d.
However, above
a
diameter
of
about
900
or
1200
mm (36 or 48
in.),
there
is
little increase
in C
values
according
to
Gros
[11],
who has had
many
years
of
experience
in
measuring
C
values
in the field.
The
values
of C
listed
in the first
part
of
Table
B-5
reflect
this experience.
In
addition
to the
discussion above, there
are
other
limitations
on the
value
of C.
Values
of C
less than
100 are
only applicable
for
velocities reasonably close
to 1 m/s (3
ft/s).
At
other velocities,
the
coefficients
are
somewhat
in
error.
For
water pipes, Lamont [12]
advises
the
following:
• C
values
of 140 to 150 are
suitable
for
smooth
(or
lined) pipes larger than
300 mm
(12
in.).
• For
smaller smooth pipes,
C
varies
from
130 to 140
depending
on
diameter.
• C
values
from
100 to 150 are
applicable
in the
tran-
sitional zone (between laminar
and
turbulent
flow),
but the
scale
effect
for
different
diameters
is not
included
in the
formula.
• The
formula
is
unsuitable and, hence,
not
recom-
mended
for
old, rough,
or
tuberculated pipes with
C
values
below 100.
•
Force mains
for
wastewater
can
become coated
with
grease
and C
values
may
vary down
to 120 for
severe grease deposition.
In
the
past (before 1940
or
1950),
it was
common
to
line steel
and
cast-iron pipe with
hot
coal-tar dip,
which
provided poor protection
and
allowed
C
values
to
drop
from
130
for new
pipe
to 100 or
less
for
pipe
in
service
for 20 yr or
more
[13].
The
modern
use of
cement mortar
or
plastic linings makes pipe very
smooth, prevents corrosion
and
tuberculation,
and
maintains
its
smoothness
indefinitely.
In field
mea-
surements
[14]
made
all
over
the
United States
on new
water
pipe
with
diameters
of 100 to 750 mm (4 to 30
in.)
lined
with
cement mortar,
the
values
of C
varied
from
134
to
151
(median
=
149, average
=
144).
For
150
to 900 mm (6 to 36
in.) pipe
in
service
for 12 to
39 yr, C
varied
from
125 to
151
(median
=
139, aver-
age
=
140)
—
a
decrease
of
only about
5
units.
Water
treatment
often
creates deposits that greatly
increase
friction
in
pipes.
In one
pipeline, lime incrus-
tation
reduced
the
measured value
of C to
only
80
downstream
from
the
treatment plant. Pipe can, how-
ever,
be
cleaned
and
relined with cement mortar
in
situ
and
restored nearly
to its
original smoothness.
Under some circumstances, deposits
of
bacterial slime
in
water pipes
can
change
the
smoothest pipe (what-
ever
the
material) into very rough
pipe.
Fortunately,
chlorination
destroys
the
slime
and
restores
the
former
smoothness.
In New
York,
for
example,
the C
factor
for
a
1800-mm
(72-in.) water main
7.7 km
(4.8
mi)
long drops
from
140 to
120
about twice
per
year
and
is
chlorinated
to
restore
the C
value
to
140. Another
example
is a
1050-mm
(42-in.) cement-lined steel cyl-
inder prestressed concrete transmission main
48 km
(30 mi)
long.
It
develops
a
slime layer only about
3
mm
(1/8 in.) thick every
five
years,
but the
thin slime
is
sufficient
to
decrease
the C
value
from
140 to
100.
A
massive dose
of
chlorine restores
its
former
smoothness. Instead
of
massive doses
of
chlorine
at
long intervals, however,
the
maintenance
of a
free
chlorine residual
of
about
0.5
mg/L
or a
stronger
(2 to
3
mg/L)
dose
for two
hours twice
per
week
in a
south-
ern
California pipeline
has
been reported
to
maintain
the
original capacity.
As
bacteria
do not
develop
immunity
to
chlorine, experimentation with doses,
contact time,
and
time intervals between doses
offers
an
opportunity
to
achieve overall economy
[15].
Bio-
fouling
is far
more prevalent than most realize,
so
chlorination facilities must
be
added
for
pipelines
sub-
ject
to
slime buildup.
The
coefficient
of
friction should
be
determined when
a new
pipeline
is first put
into
service
to
establish
an
irrefutable reference point
for
future
cleaning needs
and for
evaluating cleaning pro-
cedures.
Sewers
often
become
fouled
with grease,
and
grease
from
industries (such
as
commercial laundries,
slaughterhouses,
or
locomotive repair shops)
can
reduce
the
diameter
of
wastewater
pipes
by
one-third
or
more. Their original size
and
smoothness
can be
restored, however,
by
cleaning
the
pipe
in
place.
To
prevent excessive buildup
of
grease, include
pig
launching
and
recovery stations (see Section 4-9).
The
headloss
in
pumping station piping
is
usually
small (about
2 m or 5 ft) and is
largely related
to
valve
and
fitting
losses (see Tables
B
-6
and
B
-7),
so the
selection
of a C
value
for
piping within
the
pumping
station
is of
minor importance.
If the
static
lift
is the
major
part
of the TDH and the
transmission
or
force
main
is
short,
say
15Om
(500
ft) or
less,
the C
value
is
again
of
minor importance.
Long Force Mains
Friction
coefficients
for
long
force
or
transmission
mains must
be
established with great care. Using
the
Hazen-Williams
formula
can
lead
to
serious errors,
particularly
for (1)
large pipes,
(2)
high velocities,
or
(3)
water temperature that
differs
from
15
0
C
(6O
0
F)
by
more than about
U
0
C
(2O
0
F).
For
such situations,
use
the
Darcy-Weisbach equation.
If the
energy loss
is
a
vital design consideration, search
the
literature
for
tests
on
similar conduits instead
of
attempting
to
use
the
Moody diagram
for the
Darcy-Weisbach
equation.
Pump
and
Impeller
Selection
To
base pump operating points
on
station curves
drawn
for an
unrealistic roughness
is a
serious blun-
der.
If
some jurisdiction requires
the use of
some spe-
cific
value
of
roughness deemed
by the
designer
to be
too
great (for example,
a C
value
of
120,
as
specified
in
the
Ten-State Standards),
use
that value only
to find
the
size
of the
pipe. Select
the
pump
and its
impeller
for
a
rational envelope
of
curves that include
the
max-
imum
and
minimum limit
of
possible roughness,
for
example,
150
> C >
120.
By
choosing
a
pump that
can
accommodate impellers
of a
substantial range
of
diameters,
the
pump
can be
modified
to
operate
at its
best
efficiency
point (bep)
for any
curve within
the
envelope. However, assuming
an
excessively rough
pipe
can be
disastrous.
One
pumping station featured
several sets
of two
pumps
in
series
to
develop
the
head
calculated
for a
single
C
value
of
100.
To
keep
the
pumps
from
vibrating,
the
operators partly closed
a
downstream
valve.
A
better solution would have been
to
bypass
the
tandem pump
and
achieve
a
savings
of
$100,000
per
year
in
electric power.
It
is
wise
to use a
calibrated pressure gauge
in
mea-
suring
the
total dynamic head (TDH) during
the
start-
up
procedure
so
that
the
impeller trim can,
if
neces-
sary,
be
refined
with confidence.
3-3.
Pipe
Tables
So
many materials, pipe diameters, wall thicknesses,
and
liner thicknesses
can be
used
for
pumping sta-
tions, yard piping,
and
transmission
or
force mains
that
complete tables
of flow and
headloss would have
to
be
extensive indeed. Because interpolation between
tabular
values
is
onerous, calculating
the flow and
headloss with
the
Hazen-Williams
formula
is
much
quicker. Tables, however,
can be
used advantageously
to find
quickly
the
proper size
of
pipe,
to
approximate
the
friction
headloss,
and to
provide
an
independent
check
on a
solution
by
formula.
The
purposes
of the
pipe tables
in
Appendix
B
(Tables
B-I
to
B-4)
are the
following:
• For a
quick, preliminary determination
of
pipe size,
discharge,
and
headloss
for a
moderate friction
coefficient
(C =
120)
and
velocity
(2 m/s in
Tables
B-I
and B-3 and 5
ft/s
in
Tables
B-2 and
B-4);
• For finding the
available sizes
and
weights
of the
thinnest
(and most common) pipes used within
pumping
stations;
• For a
quick, rough check
of flow or
headloss
found
by
other means;
and
• For
providing
useful
data
for
both ductile iron pipe
(DIP)
and
steel
in
both
SI and
U.S. customary units.
For final
design,
for
different
conditions,
and for
piping outside
of the
pumping station, calculate
flow
and
headloss with
the
Darcy-Weisbach
formula
and
consult
the
tables
to
check
for
blunders.
Pumping
Station
Piping
The
joints
in
piping within
a
pumping station should
almost universally
be
bolted
flanges
augmented with
a few
strategically placed grooved-end couplings
(e.g., Victaulic®)
or
sleeve pipe couplings
(e.g.,
Dresser®)
to
permit disassembly
and
allow
for
mis-
alignment (see Chapter
4 for
illustrations). Connec-
tions
to
equipment (such
as
pumps) should
be flexible
to
prevent
the
transmission
of
undue forces, including
sheer. Mason radial
ply
joints
of
resilient material
(rubber, Viton,
or
Buna
N) are
excellent. Occasion-
ally other joints
are
used,
but
only with many
grooved-end couplings added
for
ease
in
disassembly.
Ductile iron pipe (DIP)
up to 750 mm (30
in.)
in
diameter seems
to be
preferred
by
Eastern designers
both
for
water
and
sewage pumping,
but
steel pipe
is
often
used
in
very large sizes
and in all
sizes
on the
West
Coast where
freight
makes
DIP
very expensive.
Some designers prefer
steel
pipe because
the
mitered
and
welded
fittings
allow greater
flexibility in
layout.
A 48°
bend,
for
example,
is as
easy
to
fabricate
as a
45°
bend. Steel pipe
has the
advantage
of
more con-
venient modification.
Flanges
for
steel pipe
are
always welded. Schedule
40,
according
to
ANSI
B36.10,
is
considered
to be the
"standard"
wall thickness
for
pipe
up to 600 mm (24
in.).
But for
600-mm
(24-in.)
or
larger steel pipe, most
U.S. manufacturers produce
selected
OD
cylinders
with
fractional inch plate. Deviations
from
a
pipe
maker's norms
are
expensive.
To
avoid blunders,
the
wise
designer becomes familiar with
the
sizes
of
pipe
readily available
and
with
the
many codes listed
in
Chapter
4 for
pipe
and
connections. (For example,
Schedule
40
pipe
can be
threaded
if it is
steel
but not if
it
carries
flammable fluids and
never
if it is
polyvinyl
chloride [PVC].)
In the
United States
flanges for DIP
are
always screwed
to the
barrel
by the
pipe manufac-
turer,
and
because Class
53 is the
thinnest
DIP
allowed
by
ANSI/AWWA
Cl
15/A21.
15-83
for
threaded
flanges,
it
is
therefore
the
most common thickness
used within
a
pumping station. Grooved-end cou-
plings also require Class
53 DIP for
diameters
of 400
mm
(16
in.)
or
less,
but
even thicker pipe
is
required
for
larger sizes according
to
AWWA
C
151
and
C606.
For
yard piping
or for
transmission
or
force mains,
mechanical joints
or
push-on joints would
be
pre-
ferred,
and
Class
50 DIP or any of the
several other
pipe materials would probably
be
used.
At 2
m/s
(6.5
ft/s)
velocity,
the
discharge
in
Class
50 DIP is
greater
than
in
Class
53 by
about
2 to 6% and the
headloss
is
less
by
about
1 to 3%.
Because
the
concern here
is
mainly
with
the
pumping station proper,
the
tables
in
Appendix
B are
limited
to
standard weight
for
steel
and
Class
53 for
DIP;
see the
manufacturers'
literature
for
other piping.
Linings
Shop lining applied
to DIP is
usually
one
layer
of
cement mortar centrifugally cast with minimum
thicknesses varying
from
1.6 mm
(
1
I
16
in.)
for
small
pipe
to 3.2 mm
(
l
/
s
in.)
for
large
pipe
and
given
a
thin
asphaltic
seal coat
to
control curing
per
ANSI/
AWWA
C104/A21.4.
Optionally,
a
double thickness
can
be
specified.
The
thicknesses
of
actual linings
may
vary considerably
from
those specified. These
linings
are so
thin that many engineers specify double
thickness
to
ensure adequate coverage,
eliminate
the
danger
of
pinholes,
and
provide greater integrity.
Shop linings applied
to
steel pipe
can be
coal-tar,
enamel, thin plastic,
or
thick cement mortar varying
from
6 to 13 mm
(V
4
to
V
2
in.)
per
AWWA
C205
(see
also Table 4-6).
Field-applied cement-mortar linings, according
to
Table
4-7,
can
vary
from
3 to 13 mm
(Vg
to
l
/
2
in.).
Considering
all the
possibilities
for
pipe thickness
and
for
shop
and field
linings, inside diameters
can
vary
substantially. Designers should determine
the
metal
IDs
from
manufacturers' catalogs. Determine
the
probable
net ID and the
probable range
of
friction
coefficients
carefully.
For final
design, trust
no
table,
but
calculate
the flow by
formula.
If the
pipe
is
larger
than
about
600 mm (24
in.)
or
smaller than about
75
mm
(3
in.),
or if the
temperature
is
less than about
1O
0
C
(5O
0
F)
or
more than about
3O
0
C
(86
0
F),
do not
trust
the
Hazen-Williams
formula.
Use the
Darcy-
Weisbach formula instead.
Example
3-1
Designing
Pipe
with
the
Pipe Tables
Problem:
Select
the
pipe
for a
water pumping station with
a
15-km-
(9.3-mi-)
long transmission
main. Maximum
flow is 0.4
m
3
/s
(6360
gal/min
or
14.1
ft
3
/s).
Solution:
One
choice
for the
pumping station
is DIP
lined with cement mortar
and
sized
for a
velocity
of
about
2.5 m/s
(8.2
ft/s),
which
is
high enough
to
minimize
the
size
and
cost
of
valves
and
other
fittings and low
enough
to
prevent cavitation
and
excessive headloss.
Using
Table
B-I
for SI
units
(or
Table
B-2 for
U.S. customary units),
Sl
Units
U.S. Customary
Units
Pipe
size
for v = 2
m/s:
500 mm
Pipe size
for v = 5
ft/s:
24 in.
v
desired
_
2.5m/S
_i
ox
-
v
desired
_ 8.2 _
.^.
v
TableB-l
2m
/
S
~
v
Table
B.2
5
.
,
0.213m
2
m-7
2
A
.
,
3.32ft
2
~
n
~
f
i
Area
required
=
—
—
—
—
=
0.17
m
Area required
= =
2.02
ft
1.25
1.64
Choose
450-mm pipe:
A =
0.172
m
2
Choose
18-in.
pipe:
A =
1.85
ft
2
v
= QIA =
0.4/0.172
=
2.33
m/s v = QIA =
14.1/1.85
=
7.78 ft/s
Friction
headloss:
use
Equation 3-9a
and C = 120
Friction headloss:
use
Equation 3-9b
and C = 120
*<
=
1OJOo(MJ-V
468
)-
4
-
87
h{
=
lo.soo^)'^!^)-
4
-
87
hf
=
1
1
.3
m/1000
m
h
f
=
1 1
.3
ft/1000
ft
Air in
Pipelines
Dissolved
air (or
other
gas)
is a
serious problem
in
pipelines that have intermediate high points
or are
nearly
flat. If air
comes
out of
solution,
it
forms
bub-
bles that,
in
turn,
reduce
the
water cross-sectional area
and
increase resistance
to flow
—
sometimes
greatly
—
and
the
air-moisture environment
is
conducive
to
cor-
rosion. Various ways
to
deal with
air in
pipelines
include:
(a)
designing
the
pipeline
profile
to
rise
all
the way to the
exit,
(b)
installing
air
release valves
at
high points
in the
pipeline
(or at
frequent
intervals
for
flat
profiles),
or (c)
designing
for
velocities high
enough
to
scour
air
bubbles
to the
exit. Obviously,
the
first
is
preferred
if
possible.
Air
release
valves
are
risky
because
of
uncertain maintenance. They should
not be
used
at all on
sewage force mains, because
maintenance must
be
done
so
frequently
(for example,
Always
check such calculations with
the
pipe table. Note that headloss
is a
function
of
Q or v
to
the
1.85
power. Hence,
,
fGactualV'
85
fCactuaA
1
'
85
*f-
=
*4fi^J
**-
=
"4(S)
/04
A
1
-
85
-»GSi)
.,,(fa)-
=
11.2m/100Om
=
11.2
ft/1000
ft
Entrance,
fitting,
and
valve
losses
must
be
added
(see
Section
3-4).
At
C =
145,
the
friction headloss
is 8.4
m/1000
m,
which,
in the
short length
of
pipe
in a
pumping
station, would only
be
about
60 mm
(0.2
ft).
At
11.3
m/1000
m, the
loss
in
head would
be
only about
40 mm
(0.1
ft)
more, which
is
insignificant.
For the
transmission main,
a
velocity
of
about
2 m/s (6
ft/s) seems likely
to be
economical
when
the
cost
of
pipe, valves
and
fittings,
water hammer control methods
and
devices, installa-
tion,
and
energy
are
analyzed for,
say,
a
20-yr
period.
Using Table
B-I
or
B-2,
a
500-mm
(20-in.) pipe would
fit the
conditions.
But
flanged
joints
are not
needed
for the
transmission
main, where mechanical
or
push-on joints allow Class
50 DIP to be
used.
By
referring
to the
DIPRA handbook
[16],
the
wall thickness
is 9.1 mm
(0.36
in.).
Let us
assume
the
cement mortar
is
to be
double thickness with negligible tolerance.
The OD
values
in
Tables
B-I
and B-2 are
correct
for all
classes
of
DIP,
so
ID
=
549-2(9.1+2x2.38)
ID =
21.60
-
2[0.36
+
2(3/32)]
=
521 mm =
20.5
in
h
(
=
1OJOO(My-V
52
I)-
4
-
87
h
(
=
10
,500(^)
1
-
85
(20.
5
)-
4
-
87
=
6.69
m/lOOOm
=
6.65
ft/1000ft
Or,
using
Table
B-1,
Or,
using
Table
B-2,
_
7
,f0.4y-
85
,
s
f6360V-
85
h
<
~
7
-
5
ll2oJ
*f
=
4
'
5
U20
J
=
6.68
m/1000
m.
Good
check.
=
6.65
ft/1000
ft.
Good
check.
For 15 km
(9.3
miles)
of
pipe,
the
total headloss
at C = 120 is 6.7 x 15 = 100 m
(330
ft) vs.
82.8
mm
(232
ft) at C =
145.
The
difference
in
headloss
and
energy
use is
important
and
worthy
of
careful
study. Check
by
using
the
Darcy-Weisbach
formula;
also consider
the
maximum
and
minimum
water temperatures.
monthly)
and
without
fail.
(See Section
7-1 for an
exception.)
If the
valves
are not
maintained properly,
they
are
worse than useless, because they then engen-
der a
false
sense
of
security. Designing
for
scouring
velocities
may
result
in
excessive head losses
and
energy
needs.
The
required scouring velocities
are
given
in
Table B-9.
Some consultants customarily install manways
at
450-m
(1500-ft)
intervals
in
water pipelines equal
to
or
larger than
900 mm (36
in.)
in
diameter
to
permit
worker
entry
and
inspection
of, and
repairs
to, the
lin-
ing
and to fix
leaks.
Air
release valves
are
required
in
the
man
way
covers
to
prevent
the
accumulation
of air
under
them.
3-4.
Headlosses
in
Pipe
Fittings
Pumping
stations contain
so
many pipe transitions
(bends,
contractions)
and
appurtenances (valves,
meters) that headlosses
due to
form
resistance (turbu-
lence
at
discontinuities)
are
usually greater than
the
frictional
resistance
of the
pipe.
The
simplest
approach
to
design
is to
express
the
headlosses
in
terms
of the
velocity head,
v
2
/2g,
usually immediately
upstream
of the
transition
or
appurtenance.
The
equa-
tion
for
these
losses
is
h
=
K%-
(3-15)
Ig
in
which
K is a
headloss
coefficient
(see Appendix
B,
Tables
B
-6
and
B
-7).
The few
exceptions
to
Equation
3-15
are
noted
in the
tables.
The
headloss
coefficient,
K,
is
only
an
approxima-
tion,
and
various publications
are not
always
in
agree-
ment
and may
differ
by 25% or
more.
The
values
in
Tables
B
-6
and
B
-7
have been
carefully
selected
from
many
sources
and are
deemed
to be
reliable.
In
Equation 3-15,
K
varies with pipe size
as
noted
in
Table B-6. Furthermore, published values
are for
isolated
fittings
with
a
long
run
(for example,
20
pipe
diameters)
of
straight pipe both upstream
and
down-
stream
from
the fitting. The
headloss
is
measured
between
one
point
a
short distance upstream
from
the
fitting
and
another point
at the
downstream
end of the
piping system. This ensures symmetrical
flow
patterns.
The
difference
in
headloss with
and
without
the fitting
is
used
to
compute
K.
Headlosses
for a
series
of
widely
separated
fittings are
therefore directly additive.
Part
of the
headloss
is due to the
turbulence
within
the fitting, but
probably about
30%
(less
for
partially
closed valves)
is due to
eddying
and
turbu-
lence
in the
downstream pipe.
So if one fitting
closely follows another
(as in a
pumping station),
the
apparent
K
value
for the first fitting is,
probably,
reduced
to
about 70%.
For
example, because
K for a
90°
bend
is
0.25 (see Table B-6),
K for two 90°
bends would
be
0.50
if the
bends were separated
by,
say,
a
dozen pipe diameters.
But if the
bends were
bolted together
to
make
a
180°
bend,
K for the
entire
bend could
be figured as
0.70
x
0.25
+
0.25
=
0.43,
which
is
within
8% of the K
value
for a
180°
bend
in
Table B-6.
As
another example,
K for a 90°
bend
consisting
of
three
30°
miters
can be
determined
directly
from
Table
B-6 as
0.30
or
indirectly
by
add-
ing
reduced
K
values
for
each miter except
the
last:
thus,
0.70
(0.
10
+
0.10)
+
0.10
=
0.24—
an
error
of
20%
(one publication lists
the K for the
mitered
fit-
ting
as
0.20).
Pumps,
especially when operating
on
either side
of
their best
efficiency
point, usually cause swirling
(rotation)
in the
discharge pipe. Swirling sometimes
also occurs
in
inlets
and
suction pipes.
The
effect
of
such
swirling
is to
increase eddy formation
and
turbu-
lence; consequently,
the
headloss
in fittings can be
doubled
or
even
tripled.
If
swirling
is
likely
to
occur
and
if
headloss within
the
pumping station
is
critical
(which
is
often
true
in
suction piping),
the
safe
and
conservative practice would
be to
design
for
headloss
without
swirling
and
again
for
headloss using, say,
200%
of the fitting
losses. Because there
is no
defini-
tive
body
of
literature about this complex subject,
designers must either rely
on
experience
or
guess
at
headlosses.
Another
method
for
computing headlosses
is to use
an
"equivalent
length"
of
straight pipe. This method
is
less accurate partly because,
for
example,
50
pipe diam-
eters
of a
smooth pipe length would have less headloss
than
the
same length
of
rough
pipe.
But,
of
course,
cor-
rections
can be
made
by
multiplying
a
tabular value
of
equivalent
length
by the
ratio
of
C
acmal
/C
tobular
.
This
method
is
used
in
pipe network analysis
for
simplifica-
tion,
but
there
is no
reason
for
using
it in
pumping sta-
tion
calculations.
3-5.
Friction
Losses
in
Open
Channel
Flow
The
most common equation used
in the
United States
for
open channel
flow is the
Manning equation.
In SI
units,
v
=
i*
2/
Y
/2
(3-16a)
where
v is
velocity
in
meters
per
second,
n is
Man-
ning's
friction
coefficient
(given
in
Appendix
B,
Table
B-5),
R is the
hydraulic radius
in
meters,
and S is the
slope
in
meters
per
meter.
In
U.S.
units,
v
=
'-
48
V
3
S
1
'
2
(3-16b)
n
where
v is
velocity
in
feet
per
second,
R is
hydraulic radius
in
feet,
S is
slope
in
feet
per
foot,
and n is
Manning's
fric-
tion
coefficient
(in
Table
B-5);
the
constant, 1.486,
con-
verts
SI to
U.S.
customary units.
The
Manning equation
can be
used
for
pipes
flowing
full,
but has no
advantage
over
the
Hazen-Williams
equation.
For
pipes
flowing
full
and
under pressure,
the
relationship between
C and n is
0.037
"
=
!'
12
^
04
(
3
-
17a
>
in
SI
units where
D is the ID in
meters.
In
U.S. cus-
tomary
units,
the
equation
is
0.037
n
=
1.07^-ooi
(3-17b)
CS°.04
where
D is the ID in
feet.
(See
Brater
and
King
[17]
for
a
more extensive discussion.)
Error
in the
Manning
Equation
In
spite
of its
common
use for
circular pipes
flowing
partly
full,
the
Manning equation
is
valid only
for
full
pipes.
In
extensive studies.
Yarnell
and
Woodward
[18]
confirmed
the
exponents
of
2
/
3
and
l
/
2
respec-
tively
for R and S in
Equation
3-16
for
full
pipes
and
also proved that
the
apparent values
of n
vary consid-
erably with depth
of flow. At
depths
from
15
to
55%
of
the
pipe diameter,
the
observed apparent
n is
about
25%
greater than
n for a
full
pipe.
Camp
[19]
added
the
work
of
Wilcox
[20]
to
that
of
Yarnell
and
Wood-
ward
to
obtain
the
curves shown
in
Figure
B-5.
(Camp
also included
a
graph
of
slopes
required
for
keeping
sewers clean
at
minimum discharge.)
Escritt
[21]
stated that
the
Manning equation
was
accurate within
a few
percent
if
half
of the
width
of
the
free
water surface were added
to the
wetted perim-
eter,
P, of the
pipe when computing
a
modified
hydraulic radius,
R
m
.
He
also proposed
an
equation
that Saatci
[22]
transformed into
the
expression
6-
sinO
*m
=
*
f
Q
.
0
(3-18)
0+
sin-
where
R
m
is the
modified
hydraulic radius
for a
given
depth,
R
f
is
hydraulic radius
(D/4)
for a
full
pipe,
and q
is
the
central angle
in
radians.
See
Figure
B
-4
for
rela-
tions between
0,
area, perimeter,
and
depth
of flow.
Equation
3-18
fits the
curves
of
observed data
in
Fig-
ure
B-5,
usually within
an
error
of
about
3%.
Wheeler
[23] reduced
the
error
to
less than
2% by
introducing
empirical
coefficients
into Equation
3-18
as
follows:
0-sin0
»--•.*
,^
1
^
4
^
Velocities, depths,
and
water cross-sectional areas
corresponding
to the
Manning equation
are
compared
in
Table
B-8,
in
which observed values
are
closely
(within
2%)
represented
by
Equation
3-19.
The
table
makes
it
particularly easy
to find
accurate values
of
the
hydraulic elements
for a
pipe
flowing
partly
full.
Wheeler
also developed
a
computer program,
PART-
FULL®, that
was
used
to find the
conjugate
and
sequent (before
and
after
a
hydraulic jump) depths
for
the flowrates
given
in
Tables
12-2
and
12-3.
Sewers constitute
one
kind
of
open channels
in
which
stable,
uniform
flow
rarely occurs because
of
discontinui-
ties
such
as
changes
of
gradient
and
because
of
con-
stantly
changing
flow, all of
which cause long backwater
curves.
For
practical purposes, however, these
effects
are
localized,
and a
steady-state equation
is
adequate
if the
equation itself
can be
verified
by
physical measurements.
Example
3-2
Design
of a
Sewer
Pipe
Problem: Design
a
sewer pipe
to
carry
a
maximum design
flow of
123
m
3
/hr
(541
gal/min).
Select
the
pipe material
and find the
required
diameter
and
slope.
Solution: Sewer pipe
is
usually clay, concrete,
or
plastic.
Plastic
is
popular because
of its
light weight, ease
of
handling, tight joints, durability,
and
economy,
but it is flexible and
tends
to
flatten,
so
care must
be
taken
to
backfill
it
properly.
Size.
Plastic
sewer pipe
is
available
in the
following nominal diameters:
200, 250, 300, 375,
450,
525, 600, etc.
mm
(8,10,12,15,18,21,24,
etc.
in.).
Below
375 mm (15
in.),
some design-
ers
consider
a
sewer pipe
flowing
"full" when
the
water surface
is at
mid-depth.
To
pick
up
grit
3-6.
Energy
in
Pressurized
Pipe
Flow
Many
problems
in
hydraulics
can be
solved
by
equating
the
energy
at two
points along
a
pipe
or a
channel. Within
a
closed system (one inlet
and one
outlet)
the
total energy
at
point
1
equals
the
total energy
at
point
2
plus
any
losses
due to
friction,
as
shown
by
Equation
3-4.
A
typi-
cal
example
is the
Venturi
meter
in
Figure 3-4.
deposited
at low flows and to
scour
the
pipe clean,
the
velocity
at
maximum
flow
should equal
or
exceed
1.1
m/s
(3.5
ft/s).
In
large
(>
600-mm
or
24-in.) pipe, keep
the
velocity greater than
1.5 m/s (5
ft/s)
to
inhibit septicity
and
odors (see Section 23-1).
Sl
Units U.S. Customary
Units
From Table
A-Il
(Appendix
A),
(123
m
3
/hr)(2.78
x
10~
4
)
=
0.0342m
3
/s
(541
gal/min)(2.23
x
1(T
3
)
=
1.21ft
3
/s
The
area
required
is,
from
Equation
3-1,
0.0342m
3
/s
=
Axl.lm/s
1.21ft
3
/s
=
Ax3.5ft/s
A =
0.0311m
2
A =
0.346ft
2
The
area
of a
half-circle
is
7iD
2
/8,
so
nD
2
/S
=
0.0311m
2
nD
2
/S
=
0.346
D
=
0.281m
=
281mm
D =
0.939ft
=
11.3
in.
Choose
a
300-mm
pipe. Choose
a
12-in.
pipe.
Slope.
Plastic pipe
is
very smooth
but it may
become coated with grease,
and
Ten-State Stan-
dards
[1]
require
n to be
0.013,
which
is
rather rough (see Table B-5).
Because
the
pipe
is
larger than required,
the
water
surface
is
below
the
center,
so the
perime-
ter is
less than half
the
circumference. Tables
for the
area
and
perimeters
of
segments
are
avail-
able
or the
following
formula
can be
used:
A
=
ir
2
(9-sin9)
(3-20)
where
A is the
area
of the
segment,
r is the
pipe radius,
and 6 is the
central angle
(in
radians)
of
the
sector
enclosing
the
segment. Substituting,
0.0311
=
5p^l
(9-sin6)
0.346
=
^0-}
(9-sin0)
9 =
2.95
radians
by
trial
9 =
2.95
radians
by
trial
The
wetted perimeter
is
P
-
r
e
=
5^52
X
2.95
=
0.443
m
P = r9 =
i(2.95)
=
1.48
ft
R
=
AIP =
0.0311
m
2
/0.443m
=
0.0702m
R = A/P =
0.346
ft
2
/1.48ft =
0.234ft
Substituting
in
Equation
3-16a,
Substituting
in
Equation
3-16b,
Llm/s
=
^(0.07O
2
)
27
Y
72
3.5ft/s
=
I^(
0
.234)
2/
Y
/2
S
=
0.007
ImAn
S =
0.0065
ft/ft
The
discrepancy
is
caused
by
inexact conversions
of SI to
U.S. units.
The
minimum slope
from
Ten-State Standards
[1]
is
0.0022
m/m.
From
the
principles presented
in
Section
3-1,
the
pressure
at
point
1,
as
measured
by a
gauge
or a
pie-
zometer,
is
greater than
at
point
2 by the
amount
Ah
a
=
A(p/y)
if the
meter
is
horizontal.
The
differential pres-
sure
can be
measured
by a
differential pressure gauge
(not
shown),
by a
mercury manometer,
or by an air
manometer.
Differential
pressure gauges
or
transduc-
ers are
costly
and
delicate
but
necessary
if a
remote
reading
is
needed. Mercury manometers
are
still
made,
but
mercury
is
poisonous
and an
accidental
spill, sure
to
occur eventually,
is
nearly impossible
to
clean
up. The air
manometer
is
sensitive, accurate
(if
purged
properly),
cheap,
and
excellent even when
homemade.
The
column
of
air, even
if
highly com-
pressed,
is
insignificant
in
mass,
so the
manometer
reading,
A/i
r
,
equals
the
true value,
A/i
a
,
with negligi-
ble
error.
The
connecting tubing
to any
pressure gauge
or
manometer must
be
purged
of air to
ensure accuracy.
In the air
manometer
of
Figure 3-4,
for
example, there
must
be no air
except between points
a and d, and
this
requires purging petcocks
or
bleeders
at
strategic loca-
tions.
A
means
of
introducing
the
required bubble
of
air
at the top of the
manometer
is
also required.
Figure
3-4.
Venturi
meter.
Example
3-3
Venturi
Meter
in a
Pipe
Problem: Assume
the
inlet diameter
in
Figure
3-4 to be 254 mm (10
in.)
and the
throat diameter
to
be
152
mm (6
in.)
The air
manometer reads
305 mm (12
in.).
Find
the flow, the
mercury
manometer reading,
and the
differential gauge
pressure.
Solution: Assume
the
friction loss between points
1 and 2 to be
zero (nearly true).
If the
datum
plane
is the
pipe centerline, Equation
3-4
becomes
2 2
V
1
V
9
*'
+
2i
=
**
+
2l
3-7.
Energy
in
Open
Channel
Flow
Water
in
open channels
can flow in
three
regimes:
•
subcritical
or
tranquil
flow
(waves
can
move
upstream; example: sluggish stream);
•
critical
flow
with standing waves (hydraulically
unique; example:
flow
over
a
broad crested weir);
•
supercritical
or
shooting
flow
(waves move only
downstream;
example: water
flowing
down
a dam
spillway).
Theory
To
establish which
of the
three
possible
flow
regimes
occurs, consider Equation
3-3 and
Figure
3-5
(review
Section
3-1 if
necessary).
E
S
= y
+
Yg
(3
"
3)
If
the
velocity
is
zero
and the
water
is
standing,
the
specific
energy equals
the
depth,
y, and
plots
as the
straight
line,
oa, in
Figure
3-5a.
At a
constant
dis-
charge, however,
E
8
is the
curved line represented
by
y
+
v
2
/2g.
In
tranquil
flow, the
velocity head
is
small,
but
as y
decreases,
the
velocity
(to
produce
the
same
discharge)
increases
—
and
very rapidly below
the
tran-
quil zone. Hence, there
is a
critical depth,
y
c
,
at
which
specific
energy
is
minimum.
To find it,
replace
v
2
with
(QIA)
2
,
differentiate
Equation
3-3
with respect
to y,
and
set the
resulting expression equal
to
zero; thus,
^
=
1
+
Sl
2A
-3^
=
0
dy
2g\^
dyj
Sl
Units
U.S.
Customary
Units
From Equation
3-1,
velocity
V
2
must
be
"
-
P
2
"
-
»">
»-
("
T
-
™-<
Substituting,
'.-'-»=
(
-^
,-,=,»=<-^
V
1
=
0.939m/s
V
1
=
3.09ft/s
Q
s
Av
a
(Q-^)
2
*
x
0.939
2
=
Av
=
1^
X
3.09
=
0.0476
m
3
/s
=
1.69
ft
3
/s
In
the
mercury manometer
of
Figure
3-4,
the
pressure
at f and h are
equal (because
the
eleva-
tions
are the
same
in a
static continuum).
The
differential
pressure between
g and h
equals
the
weight
of
mercury column
gh
minus
the
weight
of
water column
ef.
From
the
terms
of the
prob-
lem,
the
differential
pressure between points
1 and 2
(and, hence, between points
e and g) is 305
mm
(12
in.)
WC. The
specific
gravity
of
mercury (Hg)
is
13.6
and of
water
is
1.0,
so
305
=
/?
f
-ef(
1.0)
-[p
h
-gh(
13.6)]
12 =
p
f
-S(1.0)-[p
h
-^(13.6)]
p
f
=
/?
h
and
ef = gh
p
f
=
p
h
and ef = gh
gh
=
3057(13.6-1)
=
24.2
mm
Hg gh =
12/(13.6-1)
=
0.952
in. Hg
From Appendix Table
A-Il,
the
differential
gauge pressure
is
p
=
24.2
x
1.33
x
1O
-1
=
3.22
kPa
p =
0.952x4.91
x
10'
1
=
0.467
lb/in.
2