Page 174 - Water Engineering Hydraulics, Distribution and Treatment

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Chapter 5
Water Hydraulics, Transmission, and Appurtenances
5.2.4 Hydraulic Transients
11
Transmission lines are subjected to transient pressures when
2. Pressure tunnel
10
3. River crossing
valves are opened or closed or when pumps are started or
stopped. Water hammer and surge are among such transient
phenomena.
Water hammer is the pressure rise accompanying a sud-
den change in velocity. When velocity is decreased in this
way, energy of motion must be stored by elastic deformation
of the system. The sequence of phenomena that follows sud-
den closure of a gate, for example, is quite like what would
ensue if a long, rigid spring, traveling at uniform speed, were
suddenly stopped and held stationary at its forward end. A Cost (10 4 dollars) (c) 12 9 8 7 6 5 4 3 2 1. Low-pressure pipe Tunnel
pressure wave would travel back along the spring as it com- 1
pressed against the point of stoppage. Kinetic energy would
change to elastic energy. Then the spring would vibrate back 0 10 20 30 40 50 60
and forth. In a pipe, compression of the water and distention Head loss (ft) (h)
of the pipe wall replace the compression of the spring. The
Figure 5.21 Lagrangian optimization of conduit sections by
behavior of the pressure wave and the motion of the spring parallel tangents. Conversion factor: 1 ft = 0.3048 m.
and the water are identically described by the differential
equations for one-dimensional waves. Both systems would
where pumping is not required, controlling hydraulic factors
vibrate indefinitely, were it not for the dissipation of energy
are available heads and allowable velocities. Head require-
by internal friction.
ments include proper allowances for drawdown of reservoirs
Water hammer is held within bounds in small pipelines
and maintenance of pressure in the various parts of the com-
by operating them at moderate velocities, because the pres-
munity, under conditions of normal as well as peak demand.
sure rise in psi or kPa cannot exceed about 50 times the
Reservoir heads greater than necessary to transport water
velocity expressed in ft/s or m/s. In larger lines the pres-
at normal velocities may be turned into power when it is
sure is held down by arresting flows at a sufficiently slow
economical to do so.
rate to allow the relief wave to return to the point of control
Allowable velocities are governed by the characteris-
before pressures become excessive. If this is not practicable,
tics of the water carried and the magnitude of the hydraulic
pressure relief or surge valves are introduced.
transients. For silt-bearing waters, there are both lower and
Very large lines, 6 ft (1.8 m) or more in equivalent diam-
upper limits of velocity; for clear water, only an upper limit.
eter, operate economically at relatively high velocities. How-
The minimum velocity should prevent deposition of silt; it
ever, the cost of making them strong enough to withstand
lies in the vicinity of 2–2.5 ft/s (0.60–0.75 m/s). The maxi-
water hammer would ordinarily be prohibitive if the energy
mum velocity should not cause erosion or scour, nor should
could not be dissipated slowly in surge tanks. In its simplest
it endanger the conduit by excessive water hammer when
form, a surge tank is a standpipe at the end of the line next to
gates are closed quickly. Velocities of 4–6 ft/s (1.2–1.8 m/s)
the point of velocity control. If this control is a gate, the tank
are common, but the upper limit lies between 10 and 20 ft/s
accepts water and builds up back pressure when velocities
(3 and 6.1 m/s) for most materials of which supply conduits
are regulated downward. When demand on the line increases,
are built and for most types of water carried. Unlined canals
the surge tank supplies immediately needed water and gen-
impose greater restrictions. The size of force mains and of
erates the excess hydraulic gradient for accelerating the flow
gravity mains that include power generation is fixed by the
through the conduit. Following a change in the discharge
relative cost or value of the conduit and the cost of pumping
rate, the water level in a surge tank oscillates slowly up and
or power.
down until excess energy is dissipated by hydraulic friction
When aqueducts include more than one kind of con-
in the system.
duit, the most economical distribution of the available head
among the component classes is effected when the change
5.3 CAPACITY AND SIZE OF CONDUITS
in cost Δc for a given change in head Δh is the same
With rates of water consumption and fire demand known, for each kind. The proof for this statement is provided by
the capacity of individual supply conduits depends on their Lagrange’s method of undetermined multipliers. As shown
position in the waterworks system and the choice of the in Fig. 5.21 for three components of a conduit with an
designer for (a) a structure of full size or (b) duplicate lines allowable, or constrained, head loss H, the Lagrangian
staggered in time of construction. requirement of Δc ∕Δh =Δc ∕Δh =Δc ∕Δh is met
2
1
1
3
2
3
Minimum workable size is one controlling factor in the when parallel tangents to the three c : h curves identify, by
design of tunnels. Otherwise, size is determined by hydraulic trial, three heads h , h , and h that satisfy the constraint
3
1
2
and economic considerations. For a gravity system, that is, h + h + h = H.
1
3
2