Page 191 - Water Engineering Hydraulics, Distribution and Treatment

169
2. The increase in water pressure within the pipe:
Since the valve was closed in 2 s, but the time for the wave round-trip is 7 s, this is equivalent to sudden closure.
(ΔP) = c(Δv)
( P) = ( ∕g) c (Δv)
= (64.4∕32.2)(3,155)(6.5 − 0)
2
= 39,800 lb∕ft .
3. The tensile stress in the pipe, if the valve is closed in 2 s:
S = (ΔP) r∕t
(5.56)
t
k
2
= (39,800 lb∕ft )(24∕12 ft)∕[(3∕8)∕12 ft] 5.10 Additional Hydraulics Topics (5.53)
5
= 25.46 × 10 lb∕ft 2
2
= 17,680 lb∕in. (psi increase in pressure).
4. Discuss how the water hammer can be avoided.
The closure time should be at least 7.0 s, preferably10 times larger than 2 s, that is, 20 s.
Solution 2 (SI System):
1. The time of travel of the pressure wave upstream and back (round-trip):
t = 2L∕c (5.52)
t = 2 × 3,353 m∕(961.6m∕s).
t = 7.0s.
2. The increase in water pressure within the pipe:
Since the valve was closed in 2 s, but the time for the wave round-trip is 7 s, this is equivalent to sudden closure.
( P) = c (Δv) (5.53)
= (1,000)(961.6)(1.98 − 0)
= 1,905,000 N∕m 2
2
= 1,905 kN∕m .
3. The tensile stress in the pipe, if the valve is closed in 2 s:
S = (ΔP) r∕t k (5.56)
t
2
= (1,905,000 N∕m )(1.2192 m∕2)∕(9.525∕1,000 m)
8
= 1.22 × 10 N∕m 2
5 2
= 1.22 × 10 kN∕m increase in pressure.

4. Discuss how the water hammer can be avoided.
Same as Solution 1.

5.10.3 Impulse–Momentum Principles linear impulse = change in linear momentum
( ∑ )
Change in direction or magnitude of flow velocity of a liquid F t = M(Δv)
leads to change in fluid’s momentum. From kinetic mechan-
= M(v − v )
2
1
ics, it is known that
( ∑ )
F = Q (v − v )
2 1 (5.58)
= ( ∕g) Q (v − v )
2 1