Water Hydraulics - Mathematics and Statistics for the Environment - page 602

Free water surface

Wetted perimeter

2.0 ft

3.0 ft

FIGURE 22.6

For Example 22.14.

where
Q = Channel discharge capacity (ft 3 /sec).
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).

Recall that we defined the hydraulic radius of a channel as the ratio of the flow area to the wetted
perimeter P . In formula form, R = A / P . The new component here is n (the roughness coefficient),
and it 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 con-
crete pipes or lined channels is 0.013. The n values decrease as the channels become smoother. The
following example illustrates the application of Manning's formula for a channel with a rectangular
cross-section.

■ EXAMPLE 22.14
Problem: A rectangular drainage channel is 3 ft wide and is lined with concrete, as illustrated in
Figure 22.6. The bottom of the channel drops in elevation at a rate of 0.5 per 100 ft. What is the
discharge in the channel when the depth of water is 2 ft?

Solution: Assume that n = 0.013. Referring to Figure 22.6, we see that the cross-sectional flow area
( A ) = 3 ft × 2 ft = 6 ft 2 , and the wetted perimeter ( P ) = 2 ft + 3 ft + 2 ft = 7 ft. The hydraulic radius
( R ) = A / P = 6 ft 2 /7 ft = 0.86 ft. The slope ( S ) = 0.5/100 = 0.005. Applying Manning's formula, we get

20
0 013

.

066 . 0 5 .

Q = × × 6086 . × 0 005 . = 59 cfs

.

REFERENCES AND RECOMMENDED READING
McGhee, T.J. (1991). Water Supply and Sewerage , 2nd ed. McGraw-Hill, New York.
Nathanson, J.A. (1997). Basic Environmental Technology: Water Supply Waste Management, and Pollution
Control , 2nd ed. Prentice-Hall, Upper Saddle River, NJ.