9.7 Exercises
343
A: Efficiency decreases with increasing flow coefficient.
9.7.5. Calculate that the lift force part in the rotor work amounts to 15 % for
the Francis turbine with R = 0.75, of which the velocity triangles are represented in
Fig. 9.11. Determine the diameter ratio ( d 2 /d 1 ) with a 50 % lift force part. Note that
the diameter ratio strongly increases and that rotor flow turning increases.
A: The diameter ratio is 0.87.
9.7.6. Argue that for the Francis turbine with R = 0.65, of which the velocity
triangles are represented in Fig. 9.11, very low incidence is generated when the
flow rate is halved by closing the stator vanes. Assume that the internal efficiency is
unaffected, so that the rotor work remains the same. Remark the strong post-swirl,
which causes an efficiency drop. Does this efficiency drop increase or decrease the
incidence at the rotor inlet? How is the incidence affected if the degree of reaction
is lowered to R = 0.55?
A: The incidence is very low. The incidence increases by the efficiency drop. A
lower degree of reaction implies lower incidence.
9.7.7. Argue that for the Kaplan turbine with R = 0.85, of which the velocity
triangles are represented in Fig. 9.11, very low incidence is generated when flow
rate is halved by closing the rotor blades. Assume that the internal efficiency is
unaffected, so that the rotor work remains the same. Note the strong post-swirl
decreasing the actual efficiency. Does this efficiency drop increase or decrease the
incidence at the rotor inlet? How is the incidence affected, if the degree of reaction
increases to R = 0.90?
A: The incidence is very low. The incidence increases by the efficiency drop. A
higher degree of reaction implies lower incidence.
9.7.8. The Francis turbine rotor, represented in Fig. 9.16, is from the Karakaya power plant in Turkey. Features are H = 156.7 m, n = 150 rpm, P e = 340 MW,
d 1 = 5400 mm, b 1 = 1200 mm. Assess the suitability of the shape rendered in
Fig. 9.16. The overall efficiency η = 93.5 %.
A: The calculated specific speed is about 1. The ratio of width to diameter corresponds to Fig. 9.12.
9.7.9. The Kaplan turbine rotor, represented in Fig. 9.17, is from the Jebba
power plant in Nigeria. Features are H = 29.3 m, n = 93.75 rpm, P e = 102.7 MW,
d outer = 7100 mm. Asses the suitability of the shape rendered in Fig. 9.17. The overall efficiency η = 92 %.
A: The calculated specific speed is 2.75.
9.7.10. Argue that, with a reversible Francis type pump-turbine, as rendered in
Fig. 9.22, the rotational speed in pump operation must be larger than in turbine
operation for good efficiency in both modes, because of the slip effect. Reason with
β 1 = β 2 = 60°. Estimate the rotational speed ratio for a rotor with six blades and determination of the slip with Wiesner’s formula (Chap. 3 formula 3.21).
9.7.11. The internal efficiency of the Girard turbine studied in Exercise 6.10.7 of
Chap. 6 is only about 58 %. The main reason is the quite large kinetic energy at the
rotor outlet. The efficiency may be improved by mounting the rotor in a cylindrical tube and adding a bent diffuser in the style of Fig. 9.18. Due to suction by the
downward head and the pressure recovery in the diffuser, a pressure drop is then
343
A: Efficiency decreases with increasing flow coefficient.
9.7.5. Calculate that the lift force part in the rotor work amounts to 15 % for
the Francis turbine with R = 0.75, of which the velocity triangles are represented in
Fig. 9.11. Determine the diameter ratio ( d 2 /d 1 ) with a 50 % lift force part. Note that
the diameter ratio strongly increases and that rotor flow turning increases.
A: The diameter ratio is 0.87.
9.7.6. Argue that for the Francis turbine with R = 0.65, of which the velocity
triangles are represented in Fig. 9.11, very low incidence is generated when the
flow rate is halved by closing the stator vanes. Assume that the internal efficiency is
unaffected, so that the rotor work remains the same. Remark the strong post-swirl,
which causes an efficiency drop. Does this efficiency drop increase or decrease the
incidence at the rotor inlet? How is the incidence affected if the degree of reaction
is lowered to R = 0.55?
A: The incidence is very low. The incidence increases by the efficiency drop. A
lower degree of reaction implies lower incidence.
9.7.7. Argue that for the Kaplan turbine with R = 0.85, of which the velocity
triangles are represented in Fig. 9.11, very low incidence is generated when flow
rate is halved by closing the rotor blades. Assume that the internal efficiency is
unaffected, so that the rotor work remains the same. Note the strong post-swirl
decreasing the actual efficiency. Does this efficiency drop increase or decrease the
incidence at the rotor inlet? How is the incidence affected, if the degree of reaction
increases to R = 0.90?
A: The incidence is very low. The incidence increases by the efficiency drop. A
higher degree of reaction implies lower incidence.
9.7.8. The Francis turbine rotor, represented in Fig. 9.16, is from the Karakaya power plant in Turkey. Features are H = 156.7 m, n = 150 rpm, P e = 340 MW,
d 1 = 5400 mm, b 1 = 1200 mm. Assess the suitability of the shape rendered in
Fig. 9.16. The overall efficiency η = 93.5 %.
A: The calculated specific speed is about 1. The ratio of width to diameter corresponds to Fig. 9.12.
9.7.9. The Kaplan turbine rotor, represented in Fig. 9.17, is from the Jebba
power plant in Nigeria. Features are H = 29.3 m, n = 93.75 rpm, P e = 102.7 MW,
d outer = 7100 mm. Asses the suitability of the shape rendered in Fig. 9.17. The overall efficiency η = 92 %.
A: The calculated specific speed is 2.75.
9.7.10. Argue that, with a reversible Francis type pump-turbine, as rendered in
Fig. 9.22, the rotational speed in pump operation must be larger than in turbine
operation for good efficiency in both modes, because of the slip effect. Reason with
β 1 = β 2 = 60°. Estimate the rotational speed ratio for a rotor with six blades and determination of the slip with Wiesner’s formula (Chap. 3 formula 3.21).
9.7.11. The internal efficiency of the Girard turbine studied in Exercise 6.10.7 of
Chap. 6 is only about 58 %. The main reason is the quite large kinetic energy at the
rotor outlet. The efficiency may be improved by mounting the rotor in a cylindrical tube and adding a bent diffuser in the style of Fig. 9.18. Due to suction by the
downward head and the pressure recovery in the diffuser, a pressure drop is then