Mechanics of Fluids

622 Fluid machines
Fig. 13.20
Cavitation is a phenomenon by no means confined to turbines. Wherever
it exists it is an additional factor to be considered if dynamic similarity is
sought between one situation and another. Similarity of cavitation requires
the cavitation number (p − pv)/ 1
2ϱu2 (see Section 5.5.3) to be the same
at corresponding points. Experiments suggest, however, that similarity of
cavitation is difficult to achieve.
Example 13.2 A vertical-shaft Francis water turbine rotates at 6.25
revolutions per second under a net head of 27.5 m. The runner has,
at inlet, a diameter of 0.75 m, a flow area of 0.2 m 2 , and a blade
angle of 75 ◦ . The guide-vane angle (at outlet) is 15 ◦ . At the entrance
to the draft tube there is no whirl velocity and the absolute pressure is
35 kPa. Atmospheric pressure is 101.3 kPa. To design the draft tube,
the critical cavitation parameter σc, based on the minimum pressure in
the draft tube, is related to the power specific speed �P of the machine
by σc = 0.119(�P) 1.84 , where �P is in radians. Calculate the overall
efficiency of the turbine assuming that the mechanical efficiency is 0.97
and determine the limiting value for the height of the draft tube above
the surface of the tail race.
Solution
u1 = wr = πND = π6.25 × 0.75 m · s −1 = 14.73 m · s −1
v1 = u1 sin 105 ◦ / sin 60 ◦ = 16.42 m · s −1
vr1 = v1 sin 15 ◦ = 4.25 m · s −1
vw1 = v1 cos 15 ◦ = 15.87 m · s −1
vw2 = 0