Mechanics of Fluids

658 Fluid machines
0.85. Show that the power specific speed is 1.21 d/D rad,
where d represents the jet diameter and D the wheel diameter.
For the nozzle Cv = 0.97.
13.5 The following data refer to a Pelton wheel. Maximum overall
efficiency 79%, occurring at a speed ratio of 0.46; Cv
for nozzle = 0.97; jet turned through 165 ◦ . Assuming that
the optimum speed ratio differs from 0.5 solely as a result of
losses to windage and bearing friction which are proportional
to the square of the rotational speed, obtain a formula for the
optimum speed ratio and hence estimate the ratio of the relative
velocity at outlet from the buckets to the relative velocity
at inlet.
13.6 In a vertical-shaft inward-flow reaction turbine the sum of
the pressure and kinetic heads at entrance to the spiral casing
is 120 m and the vertical distance between this section
and the tail-race level is 3 m. The peripheral velocity of
the runner at entry is 30 m · s −1 , the radial velocity of the
water is constant at 9 m · s −1 and discharge from the runner
is without whirl. The estimated hydraulic losses are:
(1) between turbine entrance and exit from the guide vanes,
4.8 m (2) in the runner, 8.8 m, (3) in the draft tube, 790 mm,
(4) kinetic head rejected to the tail race, 460 mm. Calculate
the guide vane angle and the runner blade angle at
inlet and the pressure heads at entry to and exit from the
runner.
13.7 An inward-flow reaction turbine has an inlet guide vane angle
of 30 ◦ and the inlet edges of the runner blades are at 120 ◦ to
the direction of whirl. The breadth of the runner at inlet is
a quarter of the diameter at inlet and there is no velocity of
whirl at outlet. The overall head is 15 m and the rotational
speed 104.7 rad · s −1 (16.67 rev/s). The hydraulic and overall
efficiencies may be assumed to be 88% and 85% respectively.
Calculate the runner diameter at inlet and the power
developed. (The thickness of the blades may be neglected.)
13.8 A vertical-shaft Francis turbine, with an overall efficiency of
90%, runs at 44.86 rad · s −1 (7.14 rev/s) with a water discharge
of 15.5 m 3 · s −1 . The velocity at the inlet of the spiral
casing is 8.5 m · s −1 and the pressure head at this point is
240 m, the centre-line of the casing inlet being 3 m above the
tail-water level. The diameter of the runner at inlet is 2.23 m
and the width at inlet is 300 mm. The hydraulic efficiency is
93%. Determine (a) the output power, (b) the power specific
speed, (c) the guide vane angle, (d) the runner blade angle
at inlet, (e) the percentage of the net head which is kinetic at
entry to the runner. Assume that there is no whirl at outlet
from the runner and neglect the thickness of the blades.
13.9 The runner of a vertical-shaft Francis turbine is 450 mm diameter
and 50 mm wide at inlet and 300 mm diameter and