9 Hydraulic Turbines
344
created over the rotor. Proper functioning thus requires mounting a sealing between
the shroud of the rotor and the cylindrical tube.
Analyse the transformed machine with an unaltered stator. This means that the
pressure at the outlet of the stator (or inlet of the rotor) remains atmospheric pressure and that the outlet velocity of the stator does not change. Assume a pressure
recovery coefficient of the diffuser equal to 0.60. Consider again the operating point
with exact axial outlet of the rotor.
a. Determine the expression for the pressure at the outlet of the rotor (= inlet of the
diffuser). Due to the suction by the downward head and the pressure recovery
in the diffuser, this pressure is lower than atmospheric pressure. The pressure
difference may be expressed in energy measure by (
) / .
p
p
a − 2 ρ Calculate this
value.
b. Determine the blade speed such that the absolute velocity at outlet of the rotor is
in the axial direction. This computation requires iteration for the loss coefficient
of the rotor. Determine the loss coefficient with Soderberg’s formula (Eq. 6.14 in
Chap. 6) with aspect ratio equal to 1. A starting value may be determined from
a computation ignoring rotor loss. For the pressure difference over the rotor, the
height difference of 0.30 m between inlet and outlet of the rotor has to be taken
into account.
c. Determine the rotor work and the internal efficiency. The head supplied to the
machine is still 3.50 m. Observe the much improved efficiency.
d. Split the rotor work into the action and reaction parts. Determine the degree of
reaction. Observe that it is now much higher than with the Girard turbine.
e. Determine the speed ratio. Calculate the spouting velocity from the supplied
head of 3.50 m.
f. Make the balance of the rotor work, the losses in stator, rotor and diffuser. The
sum of the energy terms should be equal to the supplied mechanical energy.
Observe that, with respect to the Girard turbine, the rotor loss has increased
but that the diffuser loss is much lower than the sum of the losses due to outlet
kinetic energy and downward head.
A: 1a
1u
i
v
3.575m / s, v
6.191m / s, u 4.065m / s, W 25.168J / kg,
0.733, R 0.239,
0.491.
D
h
l
=
=
=
=
=
=
=
9.7.12. Results of the previous exercise are that the efficiency improves much
by adding the diffuser and that the speed ratio increases due to the increased degree
of reaction. The speed ratio may be increased further by opening the stator vanes.
Analyse the effect of opening the vanes from α 1 60
=
°
to α 1 45
=
°
and α 1 30
=
°
.
Keep the axial velocity. The consequence is that the flow rate is unchanged.
a. Determine axial and tangential components of the absolute velocity at the outlet
of the stator vanes.
b. Determine the expression for the pressure at the outlet of the stator. This pressure
is higher than atmospheric pressure. The pressure difference may be expressed in
energy measure by (
)/ .
p p a
1 −
ρ Calculate this value.
344
created over the rotor. Proper functioning thus requires mounting a sealing between
the shroud of the rotor and the cylindrical tube.
Analyse the transformed machine with an unaltered stator. This means that the
pressure at the outlet of the stator (or inlet of the rotor) remains atmospheric pressure and that the outlet velocity of the stator does not change. Assume a pressure
recovery coefficient of the diffuser equal to 0.60. Consider again the operating point
with exact axial outlet of the rotor.
a. Determine the expression for the pressure at the outlet of the rotor (= inlet of the
diffuser). Due to the suction by the downward head and the pressure recovery
in the diffuser, this pressure is lower than atmospheric pressure. The pressure
difference may be expressed in energy measure by (
) / .
p
p
a − 2 ρ Calculate this
value.
b. Determine the blade speed such that the absolute velocity at outlet of the rotor is
in the axial direction. This computation requires iteration for the loss coefficient
of the rotor. Determine the loss coefficient with Soderberg’s formula (Eq. 6.14 in
Chap. 6) with aspect ratio equal to 1. A starting value may be determined from
a computation ignoring rotor loss. For the pressure difference over the rotor, the
height difference of 0.30 m between inlet and outlet of the rotor has to be taken
into account.
c. Determine the rotor work and the internal efficiency. The head supplied to the
machine is still 3.50 m. Observe the much improved efficiency.
d. Split the rotor work into the action and reaction parts. Determine the degree of
reaction. Observe that it is now much higher than with the Girard turbine.
e. Determine the speed ratio. Calculate the spouting velocity from the supplied
head of 3.50 m.
f. Make the balance of the rotor work, the losses in stator, rotor and diffuser. The
sum of the energy terms should be equal to the supplied mechanical energy.
Observe that, with respect to the Girard turbine, the rotor loss has increased
but that the diffuser loss is much lower than the sum of the losses due to outlet
kinetic energy and downward head.
A: 1a
1u
i
v
3.575m / s, v
6.191m / s, u 4.065m / s, W 25.168J / kg,
0.733, R 0.239,
0.491.
D
h
l
=
=
=
=
=
=
=
9.7.12. Results of the previous exercise are that the efficiency improves much
by adding the diffuser and that the speed ratio increases due to the increased degree
of reaction. The speed ratio may be increased further by opening the stator vanes.
Analyse the effect of opening the vanes from α 1 60
=
°
to α 1 45
=
°
and α 1 30
=
°
.
Keep the axial velocity. The consequence is that the flow rate is unchanged.
a. Determine axial and tangential components of the absolute velocity at the outlet
of the stator vanes.
b. Determine the expression for the pressure at the outlet of the stator. This pressure
is higher than atmospheric pressure. The pressure difference may be expressed in
energy measure by (
)/ .
p p a
1 −
ρ Calculate this value.