Page 537 - 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 495

Problems/Questions 495
When a circular pipe is used, r 0.25 D, the following working s slope (dimensionless)
equations can further be derived. A>P w r hydraulic radius (m).
2
s 2.86 (v n) D 1.333 Derive the alternative Manning equations in the SI System for a
2
4.6417 (Q n) D 5.3334 square conduit, with a width of W and length equals to its width.
2
H f 2.86 L (v n) D 1.333 13.14 What will be the Manning equations in the SI System
for a rectangular conduit with dimensions of L W where
5.3334
2
4.6417 L (Q n) D
L length in m and W width in m?
where 13.15 What will be the Manning equation in the SI System for
3
Q flow (ft /s) a circular pipe with a diameter of D in m?
2
A cross-sectional area of pipe (ft )
13.16 An environmental engineer may use the nomogram in
v velocity of flow (ft/s)
Figure 13.1, for instance, for engineering design. He or she
n coefficient of roughness
should, however, also be familiar with the design equations, es-
D inside diameter of pipe (ft)
pecially when computer analysis is involved. The Manning
s slope, or rate of grade (dimensionless)
equation shown as Eq. 13.3b may be further transformed into
H f hydraulic head loss (ft)
the following design equations in the SI System for computer-
L length of the pipe (ft).)
aided engineering design assuming circular pipes are used:
Determine the required velocity (v), the required grade (s), and 2/3 1/2
the total head loss (H f ) of a 48-in. circular pipe when it is flow- v (1/n) (0.25 D) (s)
3
ing full at a discharge rate (Q) of 100 ft /s. The coefficient of Q A v
2
roughness (n) is 0.015 and the length of the pipe (L) is 1,000 ft. A 0.785 (D )
13.12 The following are two general equations for illustration where
of (a) the relationship between the flow rate in pipe 1 where 3
Q flow (m /s)
n n 1 and the flow rate in pipe 2 where n n 2 and (b) the 2
A cross-sectional area of pipe (m )
relationship between the velocity in pipe 1 where n n 1 and
v velocity of flow (m/s)
the velocity in pipe 2 where n n 2 .
n coefficient of roughness
Q 2 Q 1 (n 1 >n 2 ) D inside diameter of pipe (m)
S slope, or rate of grade (dimensionless).
v 2 v 1 (n 1 >n 2 )
Derive Eq. 13.5b from Eq. 13.3b.
where
3
3
Q 2 flow rate in pipe 2 (ft /s or m /s), when the coeffi- Given a 30.48-cm circular sewer, n 0.013, laid on a grade of
4.05% (m per 1,000 m), find its velocity of flow and rate of dis-
cient of roughness n 2
3
3
Q 1 flow rate in pipe 1 (ft /s or m /s), when the coeffi- charge using the preceding design equations.
cient of roughness n 1 13.17 The Manning equation, Eq. 13.3b, can be further derived
v 2 velocity in pipe 2 (ft/s or m/s), when the coefficient using the known SI System equations in Problem 13.16 and the
of roughness n 2 following equations for calculation of the head loss and hy-
v 1 velocity in pipe 1 (ft/s or m/s), when the coefficient draulic gradient of a circular sewer when it flows full:
of roughness n 1 2 2 5.3334
s 10.24557 Q n >(D )
n 1 the coefficient of roughness of pipe 1 (dimensionless)
2
2
n 2 the coefficient of roughness of pipe 2 (dimensionless). 10.24557 Q n D 5.3334
2
2
6.31725 v n D 1.333
Determine the flow rate for n 0.015 if the flow rate for 2 2 5.3334
3
n 0.013 is known to be 2.26 ft /s. Determine the velocity for H f 10.24557 Q n (L)/(D )
2
2
n 0.015 if the velocity for n 0.013 is known to be 2.89 ft/s. 10.24557 Q n (L)(D 5.3334 )
2
2
13.13 The Manning formula for the SI System can be written 6.31725 v n (L)(D 1.333 )
as follows:
s H f >L
v (1/n)(A>P w ) 2/3 (s) 1/2
where
where H f hydraulic head loss (m)
v velocity of flow (m/s) L length of the pipe (m)
3
n coefficient of roughness Q flow (m /s)
2
2
A cross-sectional area of pipe (m ) A cross-sectional area of pipe (m )
P w wetted perimeter (m) v velocity of flow (m/s)