Page 156 - Water Engineering Hydraulics, Distribution and Treatment

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Water Hydraulics, Transmission, and Appurtenances
Chapter 5
EXAMPLE 5.29 EFFECT OF A FLOATING OBJECT ON WATER DEPTH
A floating cylinder 8 cm in diameter and weighing 960 g is placed in a cylindrical container 26 cm in diameter partially full of water.
Determine the increase in the depth of water in the container due to placing the float in it.
Solution:
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A 960 g cylinder will displace 960 g of water. Since 1 g of water occupies 1 cm volume, the cylinder will displace 960 cm of water.
The change in total volume beneath the water surface ΔV equals the area of the cylindrical container A, times the change in
water level Δh,or
ΔV = AΔh
Δh =ΔV∕A
2
3
2
= 960 cm ∕[ (26) cm ] 3
= 1.81 cm.
5.2.2 Exponential Equation for Surface will give the following Manning formula using US customary
Resistance units:
Because of practical shortcomings of the Weisbach formula, v = (1.486∕n)(r) 0.67 (s) 0.5 (US customary units) (5.25)
engineers have resorted to the so-called exponential equa-
where v = velocity, ft/s; n =coefficient of roughness, dimen-
tions in flow calculations. Among them the Chezy formula
sionless; r = hydraulic radius, ft; and s = slope of energy
is the basic for all:
grade line, ft/ft.
√
v = C rs (5.18a) Equation (5.26) is the equivalent Manning formula using
SI units:
where v = average velocity, ft/s; C = coefficient; r = hydraulic
radius, which is defined as the cross-section area divided by v = (1∕n)(r) 0.67 (s) 0.5 (SI units) (5.26)
the wetted perimeter, ft; and s = slope of water surface or
where v = velocity, m/s; n = coefficient of roughness, dimen-
energy gradient:
sionless; r = hydraulic radius, m; and s = slope of energy
r = A∕P w (5.19) grade line, m/m.
For a pipe flowing full, the hydraulic radius, Eq. (5.19),
s = h ∕L (5.20) becomes
f
where r = hydraulic radius, ft or m; A = cross-section area, r = [( ∕4)D ]∕( D) = D∕4 (5.27)
2
2
ft 2 or m ; and P = wetted perimeter, ft or m; s =
w
slope of water surface, dimensionless; and L = pipe length, where r = hydraulic radius, ft or m; and D = pipe diameter,
ft or m. ft or m.
The coefficient C can be obtained by using one of the Substituting for r into Eqs. (5.25) and (5.26), the follow-
following expressions: ing Manning equations are obtained for practical engineering
designs for circular pipes flowing full:
Chezy expression: C = (8g∕f) 0.5 (5.21)
v = (0.59∕n)(D) 0.67 (s) 0.5 (US customary units) (5.28)
Manning expression: C = (1.486∕n)(r) 1∕6 (5.22)
Q = (0.46∕n)(D) 2.67 (s) 0.5 (US customary units) (5.29)
Bazin expression: C = 157.6∕(1 + mr −0.5 ) (5.23)
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where Q = flow rate, ft /s; v = velocity, ft/s; D = pipe diame-
Kutter expression: C = (41.65 + 0.00281∕s + 1.811∕n)∕ ter, ft; s = slope of energy grade line, ft/ft; and n = roughness
[1 + (n∕r 0.5 )(41.65 + 0.00281∕s)]
coefficient, dimensionless. For SI measurements:
(5.24) 0.67 0.5
v = (0.40∕n)(D) (s) (SI units) (5.30)
In the preceding expressions, f, n, and m are the friction 2.67 0.5
Q = (0.31∕n)(D) (s) (SI units) (5.31)
or roughness factors determined by hydraulic experiments
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using water only. Of the above hydraulic equations, Robert where Q = flow rate, m /s; v = velocity, m/s; D = pipe diam-
Manning’s expression is commonly used for both open chan- eter, m; s = slope of energy grade line, m/m; and n = rough-
nels and closed conduits. Combining Eqs. (5.18a) and (5.22) ness coefficient, dimensionless.