Page 505 - Fair, Geyer, and Okun's Water and wastewater engineering : water supply and wastewater removal

JWCL344_ch13_457-499.qxd 8/7/10 8:49 PM Page 463

13.3 Limiting Velocities of Flow 463
Conceptually, the drag exerted by flowing water on a channel is analogous to the fric-
tion exerted by a body sliding down an inclined plane. Because the volume of water per
unit surface of channel equals the hydraulic radius of the channel,
t grs (13.7a)
where t is the intensity of the tractive force, g the specific weight of water at the prevail-
ing temperature, r the hydraulic radius of the filled section, and s the slope of the invert or
loss of head in a unit length of channel when flow is steady and uniform and the water
2
surface parallels the invert. Substituting rs (v/c) in accordance with the Chézy equa-
tion, for example,
t g(v/c) 2 (13.7b)
and the tractive force intensity is seen to vary as the square of the velocity of flow v and in-
versely as the square of the Chézy coefficient, c.
13.3.1 Transporting Velocities

The velocity required to transport waterborne solids is derived from Eq. 13.4, with the
help of Fig. 13.2. For a layer of sediment of unit width and length, thickness t, and poros-
ity ratio ƒ , the drag force t exerted by the water at the surface of the sediment and just
causing it to slide down the inclined plane equals the frictional resistance R W sin a,
where W (g g)t(l ƒ ) is the weight of the sediment in water and a is the friction
s
angle. Accordingly, t (g g)t(1 ƒ ) sin a, and Eq. 13.4 becomes
s
t (g g)t(1 ƒ ) sin a k(g g)t (13.7c)
s
s
Here, k (1 ƒ ) sin a is an important characteristic of the sediment. For single grains,
the volume per unit area t becomes a function of the diameter of the grains d as an inverse
measure of the surface area of the individual grains exposed to drag or friction. Thus k
replaces k when d replaces t.
It follows from Eqs. 13.7a and 13.7c that the invert slope at which sewers will be self-
cleansing is
s (k >r)[(g g)>g] d (13.8a)
s
and that, in accordance with the Chézy equation,
v c[k d (g g)>g] 1/2 (13.9a)
s
where the value of c is chosen with full recognition of the presence of deposited or deposit-
ing solids and expressed, if so desired, in accordance with any other pertinent capacity or
friction factor. Examples include Eq. 13.9b.

T
R
t
W

Figure 13.2 Forces Acting on Sediment of Unit Width and Length and Thickness t. T
Is the Tractive Force Per Unit Surface Area. R, the Resisting Force, Is a Function of the
Weight of the Sediment (W) and the Friction Angle (a).