Science of Water : Concepts and Applications

76 The Science of Water: Concepts and Applications
where
r H = hydraulic radius
A = cross-sectional area of the water
P = wetted perimeter
Why is hydraulic radius important? Good question.
Probably the best way in which to answer this question is by illustration. Consider, for example,
that in open channels it is of primary importance to maintain the proper velocity. This is the case,
of course, because if velocity is not maintained then fl ow stops (theoretically). In order to maintain
velocity at a constant level, the channel slope must be adequate to overcome friction losses. As with
other fl ows, calculation of head loss at a given fl ow is necessary, and the Hazen–Williams equation
is useful (Q = 0.435Cd 2.63S 0.54 ). Keep in mind that the concept of slope has not changed. The difference?
We are now measuring, or calculating for, the physical slope of a channel (ft/ft), equivalent
to head loss.
The preceding seems logical, makes sense—but there is a problem. The problem is with the
diameter. In conduits that are not circular (grit chambers, contact basins, streams and rivers), or in
pipes only partially full (drains, wastewater gravity mains, sewers, etc.) where the cross-sectional
area of the water is not circular, there is no diameter.
If there is no diameter (and there is not), then what do we do? Another good question.
Because there is no diameter in a situation where the cross-sectional area of the water is not
circular, we must use another parameter to designate the size of the cross section, and the amount
of it that contacts the sides of the conduit. This is where the hydraulic radius (rH ) comes in. The
hydraulic radius is a measure of the effi ciency with which the conduit can transmit water. Its value
depends on pipe size and amount of fullness. Simply stated, we use the hydraulic radius to measure
how much of the water is in contact with the sides of the channel, or how much of the water is not
in contact with the sides (see Figure 3.19).
√ Note: For a circular channel fl owing either full or half-full, the hydraulic radius is (D/4).
Hydraulic radii of other channel shapes are easily calculated from the basic defi nition.
Hydraulic Depth
The hydraulic depth is the ratio of area in fl ow to the width of the channel at the fl uid surface. (Note
that another name for hydraulic depth is the hydraulic mean depth or hydraulic radius.)
where
d H = hydraulic depth
A = area in fl ow
w = width of the channel at the fl uid surface
FIGURE 3.19 Hydraulic radius.
d
H
A
w
Wetted perimeter Wetted area
(3.24)