Page 515 - 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 473

13.6 Alternate Stages and Critical Depths 473

For a rectangular channel (z 0), therefore,
3 3
d
h >D = d >D or h = c (13.22)
c
c
c
2 2
whence
2
v >2g = h = d c or v = 1gd c (13.23)
c
c
For a circular cross-section, finally:
1 1 -1 d c
h c 1 d c 4 p + sin C2A B - 1D
2
D
= c c10e f - 1 d + s (13.24)
D 8 D A B C1 - A BD
d c
d c
4 D D
Substituting values of d >D varying by tenths from 0.1 to 0.9 yields the numerical results
c
for h>D, v > 1gD, and 3q>(A1gD)4 2 shown in Appendix 12. A plot of Eq. 13.20 for circular
c
conduits is shown in Appendix 13.
The critical depth line in a closed conduit is seen to be asymptotic to the line
d>D 1.0; that is, there is neither a critical nor an alternate stage for an enclosed
conduit flowing full.

EXAMPLE 13.9 ALTERNATE STAGES AND CRITICAL DEPTHS
The use of Appendix 13, construction of which is simple and straightforward, can be exemplified
3
3
as follows. Given a discharge of 60 ft /s (1.7 m /s) in a 4-ft (1.22-m) circular sewer, find
1. The critical depth
2. The alternate stages for an energy head of 4 ft (1.22 m)
3. The lower alternate stage associated with an upper alternate stage at 0.8 depth
4. The sewer invert slope that would produce flow at near-critical depth.
Solution:
3 2 2
1. For q 60 ft >s and D 4 ft, 3q>(A1gD)4 = 360 * 4>(p * 16>14g)4 = 0.177:
From Appendix 13 read d c >D 0.59. Hence, d c 0.59 4 2.36 ft (0.72 m).
2
2. For h 4.0, h>D 1.0, and 3q>(A1gD)4 = 0.177 as in part l:
From Appendix 13, read d>D 0.42 and 0.95, or d l 0.42 4 1.7 ft (0.52 m), and
d u 0.95 4 3.8 ft (1.16 m).
2
3. For 3q>(A1gD)4 = 0.177 as in part 1 and d u >D 0.8, or d u 3.2 ft (0.98 m):
From Appendix 13, read d l >D 0.45, or d l 1.8 ft (0.55 m).
4. For N 0.013 and d c >D 0.59:
3
3
From Appendix 10, q>Q 0.52 and Q 60/0.52 115 ft >s (3.26 m >s)
3 3
For D 4 ft, Appendix 7, gives Q o 18.67 ft >s (0.53 m >s):
2
1/2
2
2
Since Q o S >N Q, S (QN>Q o ) . Therefore, s (115 0.013>18.67) 0.64 10 .
This is less than a 1% grade. Because many streets have slopes in excess of this, large sewers
not infrequently operate at the lower alternate depth, or high-velocity stage.