9 Hydraulic Turbines
342
blades than would be typical with a Francis turbine. The following disadvantages
of a two-machine plant may be mentioned. Pump and turbine efficiency are lower
than with a three-machine plant. Transition from pump to turbine operation or vice
versa requires inversing the sense of rotation, causing an interruption of the operation. Due to the slip effect, optimum pump operation requires a higher rotational
speed than optimum turbine operation. This is traditionally achieved by pole change
in the motor-generator. In modern plants, an asynchronous machine with variable
speed is often applied. This is a so-called doubly fed machine with a wound rotor connected to the grid through a frequency converter (see next chapter on wind
turbines). Variable rotational speed also improves the efficiency at a variable level
difference between the reservoirs. This increases the pump-turbine cycle efficiency
to about 80 %, where without variable speed it is around 75 %.
9.7 Exercises
9.7.1. The Pelton turbine rotor, shown in Fig. 9.8, is from the San Carlos hydraulic plant in Colombia. The features are: head 587 m, power per turbine 175 MW,
rotational speed 300 rpm, (outer) rotor diameter 4100 mm, six injectors per rotor.
Assess the suitability of these features, given the overall efficiency η = 91 %.
A: The diameter corresponding to the optimal speed ratio is 3.18 m, being 77.5 %
of the outer diameter. This seems good. Specific speed per injector is 0.11.
9.7.2. Calculate the internal efficiency of the Francis turbines of which the velocity triangles are sketched in Fig. 9.11. Take as loss coefficient for the diffuser
ξ d = 0.25 and determine the loss coefficient of the distributor and the rotor from
Soderberg’s formula for aspect ratio 4: ξ
δ
=
+
°
0 045 1
9 0
2
. ( ( / ) ), with δ
°
the flow
turning in degrees. Study the distribution of the losses, especially the value of rotor
losses compared to diffuser losses. Compare efficiency values to those rendered in
Fig. 9.13.
A: Diffuser losses are dominant at a low degree of reaction (low specific speed).
With a high degree of reaction (high specific speed), rotor losses exceed diffuser
losses (remarkable!). Calculated efficiency corresponds to Fig. 9.13 for specific
speed above two. Calculations overestimate the efficiency at low specific speed.
Friction losses in stator and rotor are underestimated by taking the aspect ratio equal
to four because of the narrow distributor and rotor channels. Results strongly improve by taking the correct aspect ratio into account, according to formula 6.14 in
Chap. 6.
9.7.3. Argue, by extrapolating the kinematic features rendered in Fig. 9.11, that a
Francis turbine with R = 0.50 is possible. Note that this requires a small ratio of the
outlet diameter to the inlet diameter. Draw the velocity triangles.
9.7.4. Argue, starting from the velocity triangles rendered for the Kaplan turbine in Fig. 9.11, how efficiency evolves when the flow coefficient (
/ )
φ = v
u
a
2
1
increases. Velocity triangles are sketched for ϕ = 0.30. Reason for ϕ = 0.30, ϕ = 0.40,
ϕ = 0.50. Calculate loss coefficients with Soderberg’s formula.
342
blades than would be typical with a Francis turbine. The following disadvantages
of a two-machine plant may be mentioned. Pump and turbine efficiency are lower
than with a three-machine plant. Transition from pump to turbine operation or vice
versa requires inversing the sense of rotation, causing an interruption of the operation. Due to the slip effect, optimum pump operation requires a higher rotational
speed than optimum turbine operation. This is traditionally achieved by pole change
in the motor-generator. In modern plants, an asynchronous machine with variable
speed is often applied. This is a so-called doubly fed machine with a wound rotor connected to the grid through a frequency converter (see next chapter on wind
turbines). Variable rotational speed also improves the efficiency at a variable level
difference between the reservoirs. This increases the pump-turbine cycle efficiency
to about 80 %, where without variable speed it is around 75 %.
9.7 Exercises
9.7.1. The Pelton turbine rotor, shown in Fig. 9.8, is from the San Carlos hydraulic plant in Colombia. The features are: head 587 m, power per turbine 175 MW,
rotational speed 300 rpm, (outer) rotor diameter 4100 mm, six injectors per rotor.
Assess the suitability of these features, given the overall efficiency η = 91 %.
A: The diameter corresponding to the optimal speed ratio is 3.18 m, being 77.5 %
of the outer diameter. This seems good. Specific speed per injector is 0.11.
9.7.2. Calculate the internal efficiency of the Francis turbines of which the velocity triangles are sketched in Fig. 9.11. Take as loss coefficient for the diffuser
ξ d = 0.25 and determine the loss coefficient of the distributor and the rotor from
Soderberg’s formula for aspect ratio 4: ξ
δ
=
+
°
0 045 1
9 0
2
. ( ( / ) ), with δ
°
the flow
turning in degrees. Study the distribution of the losses, especially the value of rotor
losses compared to diffuser losses. Compare efficiency values to those rendered in
Fig. 9.13.
A: Diffuser losses are dominant at a low degree of reaction (low specific speed).
With a high degree of reaction (high specific speed), rotor losses exceed diffuser
losses (remarkable!). Calculated efficiency corresponds to Fig. 9.13 for specific
speed above two. Calculations overestimate the efficiency at low specific speed.
Friction losses in stator and rotor are underestimated by taking the aspect ratio equal
to four because of the narrow distributor and rotor channels. Results strongly improve by taking the correct aspect ratio into account, according to formula 6.14 in
Chap. 6.
9.7.3. Argue, by extrapolating the kinematic features rendered in Fig. 9.11, that a
Francis turbine with R = 0.50 is possible. Note that this requires a small ratio of the
outlet diameter to the inlet diameter. Draw the velocity triangles.
9.7.4. Argue, starting from the velocity triangles rendered for the Kaplan turbine in Fig. 9.11, how efficiency evolves when the flow coefficient (
/ )
φ = v
u
a
2
1
increases. Velocity triangles are sketched for ϕ = 0.30. Reason for ϕ = 0.30, ϕ = 0.40,
ϕ = 0.50. Calculate loss coefficients with Soderberg’s formula.