At a discharge measuring station a cross-section is measured as shown below.
a) Compute the conveyance of the channel for levels h=2.00 m and h=3.00 m, respectively. The Manning numbers vary along the cross-section and their values have been indicated in Figure 6.8..
b) Compute discharges for both levels given an approximate bed slope I0=10-3.
c) Compute approximate flood wave celerities for the given water levels. Explain the difference in the celerities.
d) During the passage of a flood wave, the water level rises from 2.00 m to 3.00 m above the reference level over a period of 10 minutes. Compute the approximate deviation between the water level slope and the bed slope. (Note: transform the time to a length via the wave celerity).
e) Compute the approximate difference in percentage between the expected measured discharges and the discharges based on a rating curve.
7
Water Hammer
7.1
Introduction
Water hammer is the phenomenon of high-amplitude pressure waves travelling in pipes, caused by the rapid changes in the velocity of the transported fluid. These variations usually result from the operation of flow- and pressure control devices, such as valves, pumps and turbines.
D
dslot
Figure 7.1 Sketch of pipe with artificial slot at the top to represent storage effects of pipe expansion and water compressibility
In principle, the equations used for the computation of these pressure waves are the same as those derived for open channel flow. As an analogy one may consider a pipe which has an artificial slot at the top, in which a free surface water level may rise after complete filling of the pipe (Figure 7.1).
The wetted area of the slot represents the storage of water resulting from changes in the water pressure. The storage capacity consists of three principal contributions: 1. the elasticity of the pipe wall which leads to a pipe cross-section expansion at
increasing water pressure. It is evident that this type of storage depends on the cross-section shape and on the wall material properties and its composition. 2. compressibility of the water, which usually is neglected in free surface flow
computations. In the case of water hammer, however, the compressibility plays a dominant role. As the free surface flow equations are based upon a volume balance, the compression of the water represents a virtual water storage;
3. compressibility of air bubbles or vacuum bubbles contained in the water; Water hammer is important in engineering for the following reasons:
1. high pressures may build up in a pipe line to the extent that it may lead to bursting of the pipe;
2. vacuum bubbles may be generated when the pressure drops to values below approximately 30% of the atmospheric pressure. The resulting cavitation is highly aggressive to the pipe material and may lead to considerable damage; 3. in other cases, air may be entrained in the fluid at moments and locations where
the pressures are low.
For these reasons, any design related to transport of fluid in pipes should be checked against the risk of water hammer damage. There are various reasons why water hammer may occur. For details, reference is made to text books, e.g. Chaudhry (1987). Typical reasons are:
· instantaneous or very fast valve closure. Many of us know this phenomenon when closing a tab at home. A wrong design of the valve may cause a sudden deceleration of the water flow and a high pressure builds up due to the transformation of momentum of the water into an impulse;
· sudden energy demand changes in a hydropower station or turbine failure; · pump failure, for example, by a power cut. At the upstream end of the pump the
pressure will increase, while at the downstream end a negative wave is generated which may cause cavitation or implosion of the pipe;
· burst of a pipe line, which in turn generates pressure surges along the pipe. In many cases it is the way of operating the system that causes the water hammer problems. In other cases there may be accidental causes. In the design of the hydraulic system such operation problems may be foreseen and water hammer may be prevented by designing anti-water hammer arrangements. Examples are:
· strict control of valve manipulations. As will be shown by the computational procedure, slow closure of valves reduce the over pressures. Slow has to be seen in relation to the length of the pipe and the celerity with which pressure surges are travelling along the pipe;
· design and construction of a surge chamber as closely as possible upstream of the turbine or pump. This surge chamber serves as an escape for the pressure wave and reduces effectively the dangerous operation time of closure;
· closed and vented air vessels, which have a function similar to surge chambers. However, in this case the surface is formed by a pressurized air chamber. These devices are used, for example, at the downstream end of sewer pumps in order to prevent excessive under pressures;
· fly wheels, which prevent the sudden change of the pump rotational speed; · by-passes with a check-valve placed in parallel to a pump. It opens during pump
rundown and supplies water to the pipe downstream of the pump;
· pressure release valves, which open when the water pressure exceeds a maximum value admitted.
For a more extensive list of options reference is made again to the literature in this field. As water hammer computations may be complex, use is often made of standard software packages, such as the WANDA system of Delft Hydraulics (www.wldelft.nl). However, simple problems may also be investigated by using