IMPACT OF A JET
Water turbines are widely used throughout the world to generate power. In the type of water turbine referred to as a Pelton† wheel, one or more water jets are directed tangentially on to vanes or buckets that are fastened to the rim of the turbine disc. The impact of the water on the vanes generates a torque on the wheel, causing it to rotate and to develop power. Although the concept is essentially simple, such turbines can generate considerable output at high efficiency. Powers in excess of 100 MW, and hydraulic efficiencies greater than 95%, are not uncommon. It may be noted that the Pelton wheel is best suited to conditions where the available head of water is great, and the flow rate is comparatively small. For example, with a head of 100 m and a flow rate of 1 m3/s, a Pelton wheel running at some 250 rev/min could be used to develop about 900 kW. The same water power would be available if the head were only 10 m and the flow were 10m3/s, but a different type of turbine would then be needed.
To predict the output of a Pelton wheel, and to determine its optimum rotational speed, we need to understand how the deflection of the jet generates a force on the buckets, and how the force is related to the rate of momentum flow in the jet. In this experiment, we measure the force generated by a jet of water striking a flat plate or a hemispherical cup, and compare the results with the computed momentum flow rate in the jet.
Description of Apparatus
Fig 11.1 shows the arrangement, in which water supplied from the Hydraulic Bench is fed to a vertical pipe terminating in a tapered nozzle. This produces a jet of water which impinges on a vane, in the form of a flat plate or a hemispherical cup. The nozzle and vane are contained within a transparent cylinder, and at the base of the cylinder there is an outlet from which the flow is directed to the measuring tank of the †
L A Pelton was an American engineer who, in the late 19th century, made extensive experiments using various shapes of buckets, with the aim of obtaining high efficiency. He devised a bucket shape which has a central splitter to divide the jet. His improved wheel was patented in 1880.
bench. As indicated in Fig 11.1, the vane is supported by a lever which carries a jockey weight, and which is restrained by a light spring. The lever may be set to a balanced position (as indicated by a tally supported from it) by placing the jockey weight at its zero position, and then adjusting the knurled nut above the spring. Any force generated by impact of the jet on the vane may now be measured by moving the jockey weight along the lever until the tally shows that it has been restored to its original balanced position.
Fig 11.1 Arrangement of Apparatus
Theory of the Experiment
The equation of momentum is discussed in Section 1.3 of Chapter 1. Consider how it applies to the case shown schematically in Fig 11.2, which shows a jet of fluid impinging on a symmetrical vane.
Fig 11.2 Sketch of jet impinging on a vane
Let the mass flow rate in the jet be m . Imagine a control volume V, bounded by a control surface S which encloses the vane as shown. The velocity with which the jet enters the control volume is u1, in the x-direction. The jet is deflected by its impingement on the vane, so that it leaves the control volume with velocity u2, inclined at an angle β2 to the x-direction. Now the pressure over the whole surface of the jet, apart from that part where it flows over the surface of the vane, is atmospheric. Therefore, neglecting the effect of gravity, the changed direction of the jet is due solely the force generated by pressure and shear stress at the vane's surface. If this force on the jet in the direction of x be denoted by Fj, then the momentum equation in the
= m ( u 2 cosβ2 − u1 )
The force F on the vane is equal and opposite to this, namely
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