Environmental Engineering Reference
In-Depth Information
2.12.1.4.3 Slope
The
slope
(
S
) in open channel equations is the slope of the energy
line. If the flow is uniform, the slope of the energy line will be parallel to
the water surface and channel bottom. In general, the slope can be cal-
culated from the Bernoulli equation as the energy loss per unit length
of channel:
d
d
h
S
=
(2.25)
l
2.12.2 open Channel flow Calculations
As mentioned, calculating the head loss at a given flow is typi-
cally accomplished by using the Hazen-Williams equation. In addition,
in open channel flow problems, although the concept of slope has not
changed, a problem again arises with the diameter. In pipes only par-
tially full where the cross-sectional area of the water is not circular, we
have no diameter. Thus, the hydraulic radius is used for these noncir-
cular areas.
The original version of the Hazen-Williams equation incorporated
the hydraulic radius. Moreover, similar versions developed by Chezy
(pronounced “Shay-zee”), Manning, and others incorporated the hydrau-
lic radius.
manning's
formula
is the most commonly used one for open
channels:
15
.
.
66
.
5
q
=×× ×
ar
s
(2.26)
n
where:
q
= channel discharge capacity (ft
3
/s).
1.5 = constant.
n
= channel roughness coefficient.
a
= cross-sectional flow area (ft
2
).
r
= hydraulic radius of the channel (ft).
S
= slope of the channel bottom (dimensionless).
The hydraulic radius of a channel is defined as the ratio of the flow
area to the wetted perimeter
P
. In formula form,
r
=
a
/
P
. The new compo-
nent is
n
(the roughness coefficient), which depends on the material and
age of a pipe or lined channel and on topographic features for a natural
streambed. It approximates roughness in open channels and can range
from a value of 0.01 for a smooth clay pipe to 0.1 for a small natural
stream. The value of
n
commonly assumed for concrete pipes or lined
channels is 0.013.
n
values decrease as the channels get smoother (see
Table 2.4). The following example illustrates the application of Manning's
formula for a channel with a rectangular cross-section.
