Problems on hydraulic turbines.

PROBLEMS

1. A generator is driven by a small, single-jet Pelton turbine designed to have a power specific speed Ωsp 5 0.20. The effective head at nozzle inlet is 120 m and the nozzle velocity coefficient is 0.985. The runner rotates at 880 rpm, the turbine overall efficiency is 88%, and

the mechanical efficiency is 96%. If the blade speed-jet speed ratio, v 5 0.47, determine

a. the shaft power output of the turbine;

b. the volume flow rate;

c. the ratio of the wheel diameter to jet diameter.

2. a. Water is to be supplied to the Pelton wheel of a hydroelectric power plant by a pipe of uniform diameter, 400 m long, from a reservoir whose surface is 200 m vertically above the nozzles. The required volume flow of water to the Pelton wheel is 30 m3/s. If the pipe skin friction loss is not to exceed 10% of the available head and f 5 0.03, determine the minimum pipe diameter.

4Values of Cp for horizontal axis wind turbines are normally found in the range 0.3-0.35. The Betz limit for Cp is 0.593.

b. You are required to select a suitable pipe diameter from the available range of stock sizes to satisfy the criteria given. The ranges of diameters (m) available are 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, and 2.8. For the diameter you have selected, determine

a. the friction head loss in the pipe;

b. the nozzle exit velocity assuming no friction losses occur in the nozzle and the water leaves the nozzle at atmospheric pressure;

c. the total power developed by the turbine assuming that its efficiency is 75% based upon the energy available at turbine inlet.

3. A multi-jet Pelton turbine with a wheel 1.47 m diameter operates under an effective head of 200 m at nozzle inlet and uses 4 m3/s of water. Tests have proved that the wheel efficiency is 88% and the velocity coefficient of each nozzle is 0.99. Assuming that the turbine operates at a blade speed to jet speed ratio of 0.47, determine

a. the wheel rotational speed;

b. the power output and the power specific speed;

c. the bucket friction coefficient given that the relative flow is deflected 165o;

d. the required number of nozzles if the ratio of the jet diameter-mean diameter of the wheel is limited to a maximum value of 0.113.

4. A four-jet Pelton turbine is supplied by a reservoir whose surface is at an elevation of 500 m above the nozzles of the turbine. The water flows through a single pipe 600 m long, 0.75 m diameter, with a friction coefficient f 5 0.0075. Each nozzle provides a jet 75 mm diameter and the nozzle velocity coefficient KN 5 0.98. The jets impinge on the buckets of the wheel

at a radius of 0.65 m and are deflected (relative to the wheel) through an angle of 160o. Fluid friction within the buckets reduces the relative velocity by 15%. The blade speed-jet speed ratio v 5 0.48 and the mechanical efficiency of the turbine is 98%. Calculate, using an iterative process, the loss of head in the pipeline and, hence, determine for the turbine

a. the speed of rotation;

b. the overall efficiency (based on the effective head);

c. the power output;

d. the percentage of the energy available at turbine inlet that is lost as kinetic energy at turbine exit.

5. A Francis turbine operates at its maximum efficiency point at η0 5 0.94, corresponding to a power specific speed of 0.9 rad. The effective head across the turbine is 160 m and the speed required for electrical generation is 750 rpm. The runner tip speed is 0.7 times the spouting velocity, the absolute flow angle at runner entry is 72o from the radial direction, and the absolute flow at runner exit is without swirl. Assuming there are no losses in the guide vanes and the mechanical efficiency is 100%, determine

a. the turbine power and the volume flow rate;

b. the runner diameter;

c. the magnitude of the tangential component of the absolute velocity at runner inlet;

d. the axial length of the runner vanes at inlet.

6. The power specific speed of a 4 MW Francis turbine is 0.8, and the hydraulic efficiency can be assumed to be 90%. The head of water supplied to the turbine is 100 m. The runner vanes are radial at inlet and their internal diameter is three-quarters of the external diameter. The meridional velocities at runner inlet and outlet are equal to 25% and 30%, respectively, of the spouting velocity. Determine

a. the rotational speed and diameter of the runner;

b. the flow angles at outlet from the guide vanes and at runner exit;

c. the widths of the runner at inlet and at exit.

Blade thickness effects can be neglected.

7. a. Review, briefly, the phenomenon of cavitation in hydraulic turbines and indicate the places where it is likely to occur. Describe the possible effects it can have upon turbine operation and the turbine’s structural integrity. What strategies can be adopted to alleviate the onset of cavitation?

b. A Francis turbine is to be designed to produce 27 MW at a shaft speed of 94 rpm under an effective head of 27.8 m. Assuming that the optimum hydraulic efficiency is 92% and

the runner tip speed-jet speed ratio is 0.69, determine

a. the power specific speed;

b. the volume flow rate;

c. the impeller diameter and blade tip speed.

c. A 1/10 scale model is to be constructed to verify the performance targets of the prototype turbine and to determine its cavitation limits. The head of water available for the model tests is 5.0 m. When tested under dynamically similar conditions as the prototype, the NPSH HS of the model is 1.35 m. Determine for the model

a. the speed and the volume flow rate;

b. the power output, corrected using Moody’s equation to allow for scale effects (assume a value for n 5 0.2);

c. the suction specific speed ΩSS.

d. The prototype turbine operates in water at 30oC when the barometric pressure is 95 kPa.

Determine the necessary depth of submergence of that part of the turbine most likely to be prone to cavitation.

8. The preliminary design of a turbine for a new hydroelectric power scheme has under consideration a vertical-shaft Francis turbine with a hydraulic power output of 200 MW under an effective head of 110 m. For this particular design, a specific speed, Ωs 5 0.9 (rad), is selected for optimum efficiency. At runner inlet the ratio of the absolute velocity to the spouting velocity is 0.77, the absolute flow angle is 68o, and the ratio of the blade speed to the spouting velocity is 0.6583. At runner outlet, the absolute flow is to be without swirl. Determine

a. the hydraulic efficiency of the rotor;

b. the rotational speed and diameter of the rotor;

c. the volume flow rate of water;

d. the axial length of the vanes at inlet.

9. A Kaplan turbine designed with a shape factor (power specific speed) of 3.0 (rad), a runner tip diameter of 4.4 m, and a hub diameter of 2.0 m operates with a net head of 20 m and a shaft speed of 150 rpm. The absolute flow at runner exit is axial. Assuming that the hydraulic efficiency is 90% and the mechanical efficiency is 99%, determine

a. the volume flow rate and shaft power output;

b. the relative flow angles at the runner inlet and outlet at the hub, at the mean radius and at the tip.

10. A hydroelectric power station is required to generate a total of 4.2 MW from a number of single-jet Pelton wheel turbines each operating at the same rotational speed of 650 rpm, at the same power output and at a power specific speed of 1.0 rev. The nozzle efficiency ηN of each turbine can be assumed to be 0.98, the overall efficiency ηo is assumed to be 0.88, and the blades speed to jet speed ratio v is to be 0.47. If the effective head HE at the entry to the nozzles is 250 m, determine

a. the number of turbines required (round up the value obtained);

b. the wheel diameter;

c. the total flow rate.

11. a. In the previous problem, the reservoir surface is 300 m above the turbine nozzles and the water is supplied to the turbines by three pipelines, each 2 km long and of constant diameter. Using Darcy’s formula, determine a suitable diameter for the pipes assuming the friction factor f 5 0.006.

b. The chief designer of the scheme decides that a single pipeline would be more economical and that its cross-sectional area would need to be equal to the total cross-sectional area of the pipelines in the previous scheme. Determine the resulting friction head loss assuming that the friction factor remains the same and that the total flow rate is unchanged.

12. Sulzer Hydro Ltd. of Zurich at one time manufactured a six-jet vertical-shaft Pelton wheel turbine with a power rating of 174.4 MW, with a runner diameter of 4.1 m, and an operating speed of 300 rpm with an effective head of 587 m. Assuming the overall efficiency is 0.90 and the nozzle efficiency is 0.99, determine

a. the power specific speed;

b. the blade speed-jet speed ratio;

c. the volume flow rate.

Considering the values shown in Figure 9.2, comment on your result.

13. A vertical axis Francis turbine has a runner diameter of 0.825 m, operates with an effective head, HE 5 6.0 m, and produces 200 kW at the shaft. The rotational speed of the runner is 250 rpm, the overall efficiency is 0.90, and the hydraulic efficiency is 0.96. If the meridional (i.e., flow) velocity of the water through the runner is constant and equal to and the exit absolute flow is without swirl, determine the vane exit angle, the inlet angle of the runner vanes, and the runner height at inlet. Evaluate the power specific speed of the turbine and decide if the data given is consistent with the stated overall efficiency.

14. a. A prototype Francis turbine is to be designed to operate at 375 rpm at a power specific speed of 0.8 (rad), with an effective head of 25 m. Assuming the overall efficiency is 92%, the mechanical efficiency is 99%, the runner tip speed to jet speed ratio is 0.68, and the flow at runner exit has zero swirl, determine

i. the shaft power developed;

ii. the volume flow rate;

iii. the impeller diameter and blade tip speed;

iv. the absolute and relative flow angles at runner inlet if the meridional velocity is constant and equal to 7.0 m/s.

b. Using Thoma’s coefficient and the data in Figure 9.21, investigate whether the turbine is likely to experience cavitation. The vertical distance between the runner and the tailrace is 2.5 m, the atmospheric pressure is 1.0 bar, and the water temperature is 20oC.

15. For the previous problem, a 1/5 scale model turbine of the prototype is to be made and tested to check that the performance targets are valid. The test facility has an available head of 3 m.

For the model, determine

a. the rotational speed and volume flow rate;

b. the power developed (uncorrected for scale).

16. A radial flow hydraulic turbine whose design is based on a power specific speed, Ωsp 5 1.707 is to produce 25 MW from a total head, HE 5 25 m. The overall turbine efficiency ηo 5 0.92, the mechanical efficiency is 0.985, and the loss in head in the nozzles is 0.5 m. The ratio of the blade tip speed to jet speed is 0.90. Assuming the meridional velocity is constant and equal to 10 m/s and there is no swirl in the runner exit flow, determine

a. the volume flow rate through the turbine;

b. the rotational speed and diameter of the runner;

c. the absolute and relative flow angles at entry to the runner.

17. An axial-flow hydraulic turbine operates with a head of 20 m at turbine entry and develops 10 MW when running at 250 rpm. The blade tip diameter is 3 m, the hub diameter is 1.25 m, and the runner design is based upon a “free vortex.” Assuming the hydraulic efficiency is 94%, the overall efficiency is 92%, and the flow at exit is entirely axial, determine the absolute and relative flow angles upstream of the runner at the hub, mean, and tip radii.

18. a. A model of a Kaplan turbine, built to a scale of 1/6 of the full-scale prototype, develops an output of 5 kW from a net head of 1.2 m of water at a rotational speed of 300 rpm and a flow rate of 0.5 m3/s. Determine the efficiency of the model.

b. By using the scaling laws, estimate the rotational speed, flow rate, and power of the prototype turbine when running with a net head of 30 m.

c. Determine the power specific speed for both the model and the prototype, corrected with

Moody’s equation. To take account of the effects of size (scale), use the Moody formula

to estimate the full-scale efficiency, ηp, and the corresponding power.

19. A Pelton wheel turbine rotates at 240 rpm, has a pitch diameter of 3.0 m, a bucket angle of 165o, and a jet diameter of 5.0 cm. If the jet velocity at nozzle exit is 60 m/s and the relative velocity leaving the buckets is 0.9 times that at entry to the buckets, determine