PROBLEMS ON HYDRAULIC TURBINES
Q 1. A Pelton wheel develops 4.5 MW under a head of 120 m at a speed of 200 rev/min. The wheel
diameter is eight times the jet diameter. Use the experimental data of Fig.1 at maximum
efficiency to determine:
(a) the required flow rate,
b) wheel diameter,
c) diameter of each jet,
d) the number of jets required, and
e) the specific speed.
Take the coefficient of velocity, Cv = 0.94.
Q 2. The available head from reservoir level to the four nozzles of a twin-runner Pelton wheel is 305
m. The length of the supply pipe, or penstock, is 3 km with a friction factor f = 0.002 and ΣKL
= 2.0. The turbine develops 10.4 MW with an efficiency of 0.85.
(a) If the head across the turbine is 95% of the available head, what is the diameter of the
penstock?
(b) If Cv = 0.98 for each nozzle, calculate the diameter of each jet.
Q 3. A water turbine is required to operate at 420 rev/min under a net head of 3 m, a discharge of
0.312 m3/s, and with an efficiency of 0.9. Tests on a 1/6-scale model running at 2000 rev/min
are to be carried out using water.
For the model turbine, determine:
(a) the head drop,
(b) flow rate,
(c) output power, and
(d) expected efficiency.
Q 4 Water for a Pelton wheel turbine flows from the head-water and through the penstock as shown
in Fig. 2 The effective friction factor for the penstock, control valves, and the like is 0.008 and
the diameter of the jet is 0.20 m.
Determine the maximum power output.
Q 5 The device shown in Fig. 3 isused to investigate the power produced by a Pelton wheel turbine.
Water supplied at a constant flowrate issues from a nozzle and strikes the turbine buckets as
indicated. The angular velocity, ω, of the turbine wheel is varied by adjusting the tension on the
Prony brake spring, thereby varying the torque, Tshaft, applied to the output shaft. This torque
can be determined from the measured force, R, needed to keep the brake arm stationary as T shaft
= ω l, where l is the moment arm of the brake force.
Experimentally determined values of ω and R are shown in the following table.
ω, rev/min
0
360
450
600
700
940
1120
1480
R (N)
11.00
8.50
8.19
7.52
6.90
5.20
3.96
0.71
(a) Use these results to plot a graph of torque as a function of the angular velocity.
(b) On other graph plot the power output, W shaft  Tshaft  , as a function of the angular velocity.
On each of these graphs plot the theoretical curves for this turbine, assuming 100%
efficiency.
Compare the experimental and theoretical results and discuss some possible reasons for any
differences between them.
Q 6. A Francis turbine has the following dimensions: r1 = 4.5 m, r2 = 2.5 m, b1 = b2 = 0.85 m,
β1 = 75o, β2 = 100o. The angular speed is 120 rev/min, and the discharge is 150 m3/s.
For no separation at the entrance to the runner, determine:
(a)
the guide vane angle,
(b)
theoretical torque,
(c)
head, and
(d)
power.
Q 7. A hydroelectric site has available head HT = 80 m and Q = 3 m3/s. It is proposed to use a
Francis Turbine unit with the operating characteristics at optimum efficiency shown in Fig. 4.
Determine:
(a) the required speed, and
(b) diameter of the machine.
Q 8. Determine the power output, the type of turbine, and the approximate speed for the installation
shown in Fig. 5.Neglect all minor losses except those existing at the valve.
Q 9. A reaction turbine installation typically includes a draft tube (Fig. 6),whose function is to
convert kinetic energy of the fluid exiting the runner into flow energy. Consider a turbine
operating at Q = 85 m3/s and HT = 31.8 m. The radius of the draft tube at the outlet of the
runner is 2.5 m and the diameter at the end of the tube is 10.0 m. Losses in the draft tube can be
neglected.
(a) For water at 20oC, what is the pressure at the runner outlet if in Fig. 8, Z2 – Z1 = 2.5 m?
(b) What is the permissible Z2 – Z1 if the allowable cavitation number is σ = 0.14?
Q 10. A landowner has constructed an elevated reservoir as shown in Fig. 8(a). She estimates that
1200 L/min is available as continuous flow and desires to install a small Francis turbine to
generate electricity for her own use.
(a) What fluid power is expected to be delivered to the turbine?
(b) If a turbine similar to the one whose characteristics are shown in Fig.8 (b) is to be installed,
what are the speed and size required for this installation?
(c) Using an appropriate empirical equation, estimate the efficiency for this installation. Wshat
power is expected to be delivered to the generator?
Q 11. A proposed design for a hydroelectric project is based on a discharge of 0.25 m3/s through the
penstock and turbine (Fig. 8).The friction factor can be assumed constant, f = 0.015, and
the minor losses are negligible.
(a) Determine the power in kilowatts that can be expected from the facility, assuming that the
turbine efficiency is 0.85.
(b) Show that the type of machine to be installed is a Francis Turbine if the desired rotational
speednis 1200 rev/min.
(c) Determine the size, actual speed, and actual power output of the selected turbine.
Q 12. Test data for the small Francis Turbine shown in Fig. 9 is given in the table below. The test
was run at a constant 10.0 m head just upstream of the turbine. The prony brake on the turbine
output shaft was adjusted to give various angular velocities, and the force on the brake arm, F,
was recorded. Use the given data to plot curves of torque as a function of angular velocity and
turbine efficiency as a function of angular velocity.
ω, rev/min
0
1000
1500
1870
2170
2350
2580
2710
Q (m3/s) F (N)
3.65E-03 11.70
3.65E-03 10.68
3.65E-03 9.88
3.51E-03 8.50
3.34E-03 6.63
2.67E-03 3.90
2.17E-03 1.50
1.93E-03 0.40
Q 13. In an inward flow reaction turbine the supply head is 12 m and maximum discharge 0.28
m3/s. External diameter = 2xinternal diameter. Velocity of flow is constant = 0.15 2 gH .
Speed = 300 rev/min. Runner vanes are radial at inlet. Determine:
(a) guide vane angle
(b) vane angle at exit for radial discharge, and
(c) width of runner at inlet and outlet.
Take hydraulic efficiency = 0.8. Vanes occupy 10 per cent of the circumference.
Q 14. Water is supplied to an axial flow turbine under a total head of 35 m. The mean diameter of
the runner is 2 m and it rotates at 145 rev/min. Water leaves the guide vane at 30o to the
direction of runner rotation and at mean radius the angle of the runner blade at outlet is 28 o. if 7
per cent of the total head is lost in the casing and guide vanes and the relative velocity is
reduced by 8 per cent due to friction in the runner, determine the blade angle at inlet (at mean
radius) and the hydraulic efficiency of the turbine.
Soln:
H net = 0.93x35 = 32.6 m
V1  (2 g H net ) = 19.62  32.6  25.3 m / s
U   D N / 60    145  2 / 60  15.2 m / s
Therefore, from the velocity triangle (next page):
V f  25.3 Sin30 o  12.65 m / s; V 1  25.3 Cos30 o  21.91m / s
W1  14.3 m / s; Tan1 
12.65
 1.88
21.9  15.2
and 1  62.2 o
Therefore, W2  0.92 14.3  13.2 m / s and from the velocity triangle at outlet V 2  3.6 m / s
Therefore,
E  (U / g )(V 1  V 2 )  15.2 18.3 / 9.81  28.4 m
and so,  h  E / H  28.4 / 35  0.811  81.1 per cent
Q 15. The single stage, axial-flow turbomachine shown in Fig. 10 involves water flow at a
volumetric flow rate of 9 m3/s. The rotor revolves at 600 rev/min. The inner and outer radii of
the annular flow path through the stage are 0.46 and 0.61 m, and β2 = 60o. The flow entering
the rotor row and leaving the stator row is axial when viewed from the stationary casing. Is this
device a turbine or a pump? Estimate the amount of power transferred to or from the fluid.