Environmental Engineering Reference
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(7) and the Reynolds number (8).
fLV
2
D
2
g
fLQ
2
D
2
gA
p
H
f
=
=
(7)
ρ
VD
μ
VD
υ
=
=
Re
(8)
In equations (7) and (8),
Q
is the volumetric flow rate,
f
is the friction
factor that characterizes the flow resistance, and
μ
dynamic viscosity of the fluid.
Experimental values of
f
have been determined for numerous pipes of varied
geometry and material; the results of which were used to create the Moody
diagram shown as Figure 9.3. The Moody diagram is a graphical representation
of the Colebrook-White equation (9).
1
√
f
=
0
.
869 ln
2
k
s
D
18
.
7
Re
√
f
1
.
74
−
+
(9)
Solving the Colebrook-White equation for
f
requires a trial-and-error or itera-
tive solution method. Swamee and Jain (1976) developed an explicit relationship
that approximates the Colebrook-White equation and Moody diagram.
0
.
25
f
=
(10)
log
Re
0.9
2
k
s
3
.
7
D
+
5
.
74
The friction factor and the subsequent friction loss of a flowing liquid depend
on whether the flow is laminar or turbulent.
Laminar flow
exists when viscous
forces are large compared to inertial forces. When inertial forces are large com-
pared to viscous forces, the flow is considered turbulent. The Reynolds number
(Re) is the ratio of inertia forces to viscous forces and is a convenient param-
eter for characterizing laminar and turbulent flow. For Re
<
2
,
000, the flow is
laminar and
f
is solely dependent on the Re (
f
64
/
Re). Most practical pipe
flow problems are in the turbulent region. The velocity of water flowing in a 1m
diameter pipe at 20
◦
C would have to be
=
2 mm/sec to be in the laminar range.
When 2
,
000
<
Re
<
4
,
000, the flow is transitioning from laminar flow to
turbulent flow and is unstable (critical zone in Figure 9.3). In this range, friction
loss calculations are difficult because it is impossible to determine a unique value
of
f
. Fortunately, most practical pipe flow problems involve Re
>
4
,
000.
When Re
>
4
,
000, the flow becomes turbulent and
f
is a function of both
Re and the relative pipe roughness (
k
s
/D
).
k
s
represents the average height
of the material roughness elements of the conduit boundary. As the level of
flow turbulence increases (increasing Re), a wholly rough turbulent condition,
as shown in Figure 9.3, is eventually reached where
f
is only dependent upon
k
s
/D
.
Figure 9.3 gives values of
k
s
for common pipe materials. Note that a range of
k
s
values, rather than one single value, is given for each material. This is because
≤
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