Page 314 - Fluid Mechanics and Thermodynamics of Turbomachinery

Hydraulic Turbines 295
exit plane of the runner and the free surface of the tailrace. The energy equation
between the exit of the runner and the tailrace can now be written as
1 2
1 2
p 3 / C c C gz gH DT D c C p a / , (9.20)
2 3 2 4
where H DT is the loss in head in the draft tube and c 4 is the exit velocity.
The hydraulic efficiency is given by
W U 2 c 2 U 3 c 3
H D D (9.21)
gH E gH E
and, if c 3 D 0, then

U 2 c 2
H D . (9.21a)
gH E
The overall efficiency is given by 0 D m H . For large machines the mechanical
losses are relatively small and m ³ 100 per cent and so 0 ³ H .
For the Francis turbine the ratio of the runner speed to the spouting velocity,
D U/c 0 , is not as critical for high efficiency operation as it is for the Pelton
turbine and, in practice, it lies within a fairly wide range, i.e. 0.6 6 6 0.9. In
most applications of Francis turbines the turbine drives an alternator and its speed
must be maintained constant. The regulation at part load operation is achieved by
varying the angle of the guide vanes. The guide vanes are pivoted and, by means of
a gearing mechanism, the setting can be adjusted to the optimum angle. However,
operation at part load causes a whirl velocity component to be set up downstream of
the runner causing a reduction in efficiency. The strength of the vortex can be such
that cavitation can occur along the axis of the draft tube (see remarks on cavitation
later in this chapter).

EXAMPLE 9.3. In a vertical-shaft Francis turbine the available head at the inlet
flange of the turbine is 150 m and the vertical distance between the runner and
the tailrace is 2.0 m. The runner tip speed is 35 m/s, the meridional velocity of the
water through the runner is constant and equal to 10.5 m/s, the flow leaves the runner
without whirl and the velocity at exit from the draft tube is 3.5 m/s. The hydraulic
energy losses estimated for the turbine are as follows:
H N D 6.0m,H R D 10.0m,H DT D 1.0m.

Determine:
(1) the pressure head (relative to the tailrace) at inlet to and at exit from the runner;
(2) the flow angles at runner inlet and at guide vane exit;
(3) the hydraulic efficiency of the turbine.

3
If the flow discharged by the turbine is 20 m /s and the power specific speed of the
turbine is 0.8 (rad), determine the speed of rotation and the diameter of the runner.
Solution. From eqn. (9.20)
p 3 p a 1 2 2
D H 3 D .c 4 c / C H DT z.
3
g 2g