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Question Bank on Hydraulic Machines (AM601)

ON DIAMENSIONAL SIMILARITY & MODEL ANALYSIS:

1. A radial flow hydraulic turbine is required to be designed to produce 30MW under a head

of 14m at a speed of 95 rpm. A geometrically similar model with an output of 40KW and

a head of 5m is to be tested under dynamically similar conditions. At what speed, must

the model be run? What is the required impeller diameter ratio between the model and

prototype and what is the volume flow rate through the model if its efficiency can be

assumed to be 90%?

[Ans. 721.4 rpm, 0.079, 0.906m3/s]

2. Specification for an axial flow coolant pump for one loop of a pressurized water nuclear

reactor:

Head=85m. Flow rate=20,000m3/hr., Speed= 1490 rpm, Diameter=1.2m, Water

density=714 kg/m3, Power=4MW (electrical).

The manufacture plants to build up a model of the above. Test conditions limit the

available electrical power to 500kW and flow to 1.5m3/s of cold water. If the model and

prototype efficiencies are assumed equal, find the head (H2/H1), Speed (N2/N1) and scale

ratio (D2/D1) of the model. Calculate the ‘dimensionless specific speed’ of the prototype

and confirm that it is identical with the model.

[Ans. 1.0, 3.3, 0.3, 2.35 radian]

3. A projected low head hydro-electric scheme is to be set up, where 283.17 m3/s of water

are available under a head of 3.66 m. Alternative schemes to use Francis turbines having

a specific speed of 467.5 or Kaplan turbines with a specific speed of 804.1 rpm(Metric

HP)0.5/m5/4 are investigated. The normal running speed is to be 50 rpm. In both the cases.

Determine the dimensionless specific speeds and compare the two proposals in so far as

the numbers of the machines are concerned. Estimate the power developed by each

machine. The units in either installation are to be of equal power and the efficiency of

each type may be assumed to be 0.9.

[Ans. 2.5 radian for Francis, 4.3 radian for Kaplan, 1716

kW per Francis turbine, 5209 kW per Kaplan turbine, 6

Francis machines, 2 Kaplan machines]

ON PELTON TURBINE:

1. A generator is to be driven by a small Pelton wheel with a head of 91.5m at inlet to the

nozzle and discharge of 0.04m3/s. The wheel rotates at 720 rpm. And the velocity co-

efficient of the nozzle is 0.98. If the efficiency of the wheel (based on the energy

available at the nozzle is 80% and the ration of bucket speed to jet speed is 0.46,

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determine the wheel –to jet- diameter ratio at the centre-line of the buckets, and the speed

of the wheel. What is the dimensionless power specific speed of the wheel?

[Ans. 14.5, 19.1 m/s, 0.082 radian]

2. A reservoir with a height of 280m. is connected to the power house of a hydroelectric

plant through three pipes each 2.5 kg long and with friction factor (f)= 0.006, in which

the head loss is not exceed 34m. it is a requirement that a total shaft output of 18MW be

developed, and to achieve this, it is decided to install a number of single-jet Pelton

wheels, each with a dimensionless specific speed not exceeding 0.23 radian. The ratio of

bucket speed to jet speed is 0.46, while the wheel speed is to be 650 rpm. If the nozzle

have a discharge coefficient (Ku1) of 0.49 and the velocity co-efficient (Kv1) of 0.96and

assuming that each wheel has an overall efficiency of 87%, find: (a) the no. of Pelton

wheels required, (b) the wheel diameter, (c) the jet nozzle diameter and (d) the diameter

of supply pipes.

[Ans. 6, 0.901m., 0.174m., 1.07m]

3. The buckets of a Pelton wheel deflect the jet through an angle of 1700, while the relative

velocity of the water is reduced by 12% due to bucket friction. Calculate the theoretical

hydraulic efficiency from the velocity triangles for a bucket to jet speed ratio of 0.47.

Under a gross head of 600m the wheel develops 1250 kW, when the loss of head due to

pipe friction between the reservoir and nozzle is 48m. The bucket-circle diameter of the

wheel is 900mm and there are two jets. The nozzle velocity coefficient is 0.98. Find the

speed of rotation of the wheel and the diameter of the nozzles if the actual hydraulic

efficiency is 0.9 times that calculated above.

[Ans. 93%, 1017 rpm, 42.3mm]

4. A twin-jet Pelton turbine has a mean bucket-circle diameter of 1.68 m. and runs at

500rpm. When the jets are 152.5 mm diameter, the available head at the nozzle is 488m.

Assuming coefficient of velocity for the nozzle as 0.98, outlet angle of bucket 150,

relative velocity for water leaving the bucket 0.88 of that of inlet; and windage and

mechanical losses 3% of the water horse-power supplied, find: (a) the water horsepower

supplied and break horsepower: (b) the force of one jet on the bucket: (c) the overall

efficiency.

Note: 1HP = 0.746 kW, 1 kgf = 9.81 N. Students may convert answers in SI units

accordingly.

[Ans. 22.740 HP; 19.300 HP; 17,000kgf; 84.7%]

5. A Pelton wheel is working under a gross head of 400m. The water is supplied the

penstock of diameter 1m. and length 4 km, from reservoir to the Pelton wheel. The co-

efficient of friction for the penstock is given as 0.008. The jet of water of diameter

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[Ans. 81.54%]

6. An inward flow radial turbine has an overall efficiency of 74%.the net head across the

turbine is 5.5mand the required power output is 125kw.the runner tangential velocity is

0.97(2gh)1/2,while the flow velocity is 0.42(2gh)1/2.if the speed of the runner is 230rpm ,

with hydraulic loss accounting for 18%of the energy available ,calculate the inlet guide

vane exit angle ,the inlet angle of the runner vane, the runner diameter at the inlet and the

height of the runner at the inlet . Assume that the discharge is radial.

[Ans. 43.40, 143.80, 0.836m, 0.287m]

7. A Francis turbine has a diameter of 1.4m,and rotates at speed 430rpm.water enter the runner

without shock with a flow velocity of 9.5m/s and leave the runner without whirl with an

absolute velocity of 79 m/s and leaves the runner without whirl with an absolute velocity of

7 m/s. the difference between the sum of the static and potential heads at entrance to the

runner and the exit from the runner is 62m. If the turbine develop 12.25 MW and has a flow

rate of 12m3/s of the water then the net head is 115m., find (a)the absolute velocity of water

at entry to the runner and the angle of inlet guide vanes ,(b)the entry angle of the runner

blade and (c)the head lost in the runner .

[Ans. 33.8m/s, 16.30, 84.60, 13.7m]

8. An inward flow reaction turbine runs at speed of 375rpm. Under an available net total head

from inlet flange to tailrace of 62m.the external diameter of the runner is 1.5m.and the

dimensionless power specific speed based on the power transfer to the runner is

0.14rev.water enter the runner without shock with a flow velocity of 7m/s. it discharge to

the tailrace with a velocity of 2m/s. the static pressure head is 2.2m.below atmospheric

pressure .assuming a hydraulic efficiency of 90%,find (a)the runner blade entry angle

,(b)the head lost in the absolute casing and guide vanes ,in the runner and in the draft tube

and (c)the draft tube entry diameter.

[Ans. 140.40, 3.27m, 0.92m, 1.8m of water , 0.88mm]

9. (a) Show that the hydraulic efficiency h for a Francis turbine having velocity of flow

through runner as constant, is given by the equation

2

1

0.5 tan

1

tan

1

tan

h

where, α =guide blade angle & θ =runner vane angle at inlet .the turbine is having radial

discharge at outlet.

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(b) If vanes are radial at inlet, show that 2

2

2 tanh

10. The following data is given for a Francis turbine: net head=70m, speed=600rpm, shaft

power=367.875 kW, overall efficiency =85%, hydraulic efficiency 95%, flow ratio=0.25,

breadth ratio=0.1, outer diameter of the runner =2×inner diameter of the runner. The

thickness of the vanes occupy10%of the circumferential area of the runner. Velocity of flow

is constant at inlet and the outlet and discharge is radial at outlet. Determine (a) guide blade

angle (b) runner vane angle at inlet and outlet (c) diameter of runner at inlet and outlet (d)

width of wheel at inlet.

[Ans. 12020’, 18057’, 0.49m, 0.245m, 49mm]

11. A reaction turbine work at 500rpm.under a head of 100m.the diameter of turbine at inlet is

100cm.and the flow area is 0.35m2.the angle made by absolute and relative velocity at inlet

are 150 and 600 with the tangential velocity .determine (a)the volume flow rate (b)the power

developed (c)efficiency ,assume whirl at outlet to be zero.

[Ans. 2.90m3/s, 2355.4kw, 82.6%]

12. A conical draft tube having inlet and outlet diameters 1m and 1.5m.discharges water at

outlet with a velocity of 2.5m/s. the total length of the draft tube is 6m.and 1.2m.of the

length of the draft tube is immersed in water .if the atmospheric draft tube is equal to 0.2into

velocity head at the outlet of the tube, find (a) pressure head at the inlet and (b) efficiency of

the draft tube.

[Ans. 4.27m. absolute, 76.3%]

ON KAPLAN TURBINE:

1. An axial flow hydraulic turbine has a net head of 23 m. across it, and, when running at a

speed of 150 rpm, develops 23MW. The blade tip and hub diameter are 4.75 m. and 2.0

m. respectively. If the hydraulic efficiency is93%and the overall efficiency85%, calculate

the inlet and outlet blade angle at the mean radius assuming axial flow at outlet.

[Ans. 156.20, 17.20]

2. A Kaplan turbines operating under a net head of 20 m. develop 16MW with an overall

efficiency of 80%. The diameter of the runner is 4.2 m. while the hub diameter is 2 m.

and the ‘dimensionless power specific speed’ is 3 radian. If the hydraulic efficiency is

90% calculate the inlet and exit angles of the runner blades at the tip and at the hub if the

flow leaving the runner is purely axial.

[Ans. 163.40, 14.50; 127.60, 28.60]

Question Bank on Hydraulic Machines (AM601)

ON DIAMENSIONAL SIMILARITY & MODEL ANALYSIS:

1. A radial flow hydraulic turbine is required to be designed to produce 30MW under a head

of 14m at a speed of 95 rpm. A geometrically similar model with an output of 40KW and

a head of 5m is to be tested under dynamically similar conditions. At what speed, must

the model be run? What is the required impeller diameter ratio between the model and

prototype and what is the volume flow rate through the model if its efficiency can be

assumed to be 90%?

[Ans. 721.4 rpm, 0.079, 0.906m3/s]

2. Specification for an axial flow coolant pump for one loop of a pressurized water nuclear

reactor:

Head=85m. Flow rate=20,000m3/hr., Speed= 1490 rpm, Diameter=1.2m, Water

density=714 kg/m3, Power=4MW (electrical).

The manufacture plants to build up a model of the above. Test conditions limit the

available electrical power to 500kW and flow to 1.5m3/s of cold water. If the model and

prototype efficiencies are assumed equal, find the head (H2/H1), Speed (N2/N1) and scale

ratio (D2/D1) of the model. Calculate the ‘dimensionless specific speed’ of the prototype

and confirm that it is identical with the model.

[Ans. 1.0, 3.3, 0.3, 2.35 radian]

3. A projected low head hydro-electric scheme is to be set up, where 283.17 m3/s of water

are available under a head of 3.66 m. Alternative schemes to use Francis turbines having

a specific speed of 467.5 or Kaplan turbines with a specific speed of 804.1 rpm(Metric

HP)0.5/m5/4 are investigated. The normal running speed is to be 50 rpm. In both the cases.

Determine the dimensionless specific speeds and compare the two proposals in so far as

the numbers of the machines are concerned. Estimate the power developed by each

machine. The units in either installation are to be of equal power and the efficiency of

each type may be assumed to be 0.9.

[Ans. 2.5 radian for Francis, 4.3 radian for Kaplan, 1716

kW per Francis turbine, 5209 kW per Kaplan turbine, 6

Francis machines, 2 Kaplan machines]

ON PELTON TURBINE:

1. A generator is to be driven by a small Pelton wheel with a head of 91.5m at inlet to the

nozzle and discharge of 0.04m3/s. The wheel rotates at 720 rpm. And the velocity co-

efficient of the nozzle is 0.98. If the efficiency of the wheel (based on the energy

available at the nozzle is 80% and the ration of bucket speed to jet speed is 0.46,

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determine the wheel –to jet- diameter ratio at the centre-line of the buckets, and the speed

of the wheel. What is the dimensionless power specific speed of the wheel?

[Ans. 14.5, 19.1 m/s, 0.082 radian]

2. A reservoir with a height of 280m. is connected to the power house of a hydroelectric

plant through three pipes each 2.5 kg long and with friction factor (f)= 0.006, in which

the head loss is not exceed 34m. it is a requirement that a total shaft output of 18MW be

developed, and to achieve this, it is decided to install a number of single-jet Pelton

wheels, each with a dimensionless specific speed not exceeding 0.23 radian. The ratio of

bucket speed to jet speed is 0.46, while the wheel speed is to be 650 rpm. If the nozzle

have a discharge coefficient (Ku1) of 0.49 and the velocity co-efficient (Kv1) of 0.96and

assuming that each wheel has an overall efficiency of 87%, find: (a) the no. of Pelton

wheels required, (b) the wheel diameter, (c) the jet nozzle diameter and (d) the diameter

of supply pipes.

[Ans. 6, 0.901m., 0.174m., 1.07m]

3. The buckets of a Pelton wheel deflect the jet through an angle of 1700, while the relative

velocity of the water is reduced by 12% due to bucket friction. Calculate the theoretical

hydraulic efficiency from the velocity triangles for a bucket to jet speed ratio of 0.47.

Under a gross head of 600m the wheel develops 1250 kW, when the loss of head due to

pipe friction between the reservoir and nozzle is 48m. The bucket-circle diameter of the

wheel is 900mm and there are two jets. The nozzle velocity coefficient is 0.98. Find the

speed of rotation of the wheel and the diameter of the nozzles if the actual hydraulic

efficiency is 0.9 times that calculated above.

[Ans. 93%, 1017 rpm, 42.3mm]

4. A twin-jet Pelton turbine has a mean bucket-circle diameter of 1.68 m. and runs at

500rpm. When the jets are 152.5 mm diameter, the available head at the nozzle is 488m.

Assuming coefficient of velocity for the nozzle as 0.98, outlet angle of bucket 150,

relative velocity for water leaving the bucket 0.88 of that of inlet; and windage and

mechanical losses 3% of the water horse-power supplied, find: (a) the water horsepower

supplied and break horsepower: (b) the force of one jet on the bucket: (c) the overall

efficiency.

Note: 1HP = 0.746 kW, 1 kgf = 9.81 N. Students may convert answers in SI units

accordingly.

[Ans. 22.740 HP; 19.300 HP; 17,000kgf; 84.7%]

5. A Pelton wheel is working under a gross head of 400m. The water is supplied the

penstock of diameter 1m. and length 4 km, from reservoir to the Pelton wheel. The co-

efficient of friction for the penstock is given as 0.008. The jet of water of diameter

Page 5

[Ans. 81.54%]

6. An inward flow radial turbine has an overall efficiency of 74%.the net head across the

turbine is 5.5mand the required power output is 125kw.the runner tangential velocity is

0.97(2gh)1/2,while the flow velocity is 0.42(2gh)1/2.if the speed of the runner is 230rpm ,

with hydraulic loss accounting for 18%of the energy available ,calculate the inlet guide

vane exit angle ,the inlet angle of the runner vane, the runner diameter at the inlet and the

height of the runner at the inlet . Assume that the discharge is radial.

[Ans. 43.40, 143.80, 0.836m, 0.287m]

7. A Francis turbine has a diameter of 1.4m,and rotates at speed 430rpm.water enter the runner

without shock with a flow velocity of 9.5m/s and leave the runner without whirl with an

absolute velocity of 79 m/s and leaves the runner without whirl with an absolute velocity of

7 m/s. the difference between the sum of the static and potential heads at entrance to the

runner and the exit from the runner is 62m. If the turbine develop 12.25 MW and has a flow

rate of 12m3/s of the water then the net head is 115m., find (a)the absolute velocity of water

at entry to the runner and the angle of inlet guide vanes ,(b)the entry angle of the runner

blade and (c)the head lost in the runner .

[Ans. 33.8m/s, 16.30, 84.60, 13.7m]

8. An inward flow reaction turbine runs at speed of 375rpm. Under an available net total head

from inlet flange to tailrace of 62m.the external diameter of the runner is 1.5m.and the

dimensionless power specific speed based on the power transfer to the runner is

0.14rev.water enter the runner without shock with a flow velocity of 7m/s. it discharge to

the tailrace with a velocity of 2m/s. the static pressure head is 2.2m.below atmospheric

pressure .assuming a hydraulic efficiency of 90%,find (a)the runner blade entry angle

,(b)the head lost in the absolute casing and guide vanes ,in the runner and in the draft tube

and (c)the draft tube entry diameter.

[Ans. 140.40, 3.27m, 0.92m, 1.8m of water , 0.88mm]

9. (a) Show that the hydraulic efficiency h for a Francis turbine having velocity of flow

through runner as constant, is given by the equation

2

1

0.5 tan

1

tan

1

tan

h

where, α =guide blade angle & θ =runner vane angle at inlet .the turbine is having radial

discharge at outlet.

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(b) If vanes are radial at inlet, show that 2

2

2 tanh

10. The following data is given for a Francis turbine: net head=70m, speed=600rpm, shaft

power=367.875 kW, overall efficiency =85%, hydraulic efficiency 95%, flow ratio=0.25,

breadth ratio=0.1, outer diameter of the runner =2×inner diameter of the runner. The

thickness of the vanes occupy10%of the circumferential area of the runner. Velocity of flow

is constant at inlet and the outlet and discharge is radial at outlet. Determine (a) guide blade

angle (b) runner vane angle at inlet and outlet (c) diameter of runner at inlet and outlet (d)

width of wheel at inlet.

[Ans. 12020’, 18057’, 0.49m, 0.245m, 49mm]

11. A reaction turbine work at 500rpm.under a head of 100m.the diameter of turbine at inlet is

100cm.and the flow area is 0.35m2.the angle made by absolute and relative velocity at inlet

are 150 and 600 with the tangential velocity .determine (a)the volume flow rate (b)the power

developed (c)efficiency ,assume whirl at outlet to be zero.

[Ans. 2.90m3/s, 2355.4kw, 82.6%]

12. A conical draft tube having inlet and outlet diameters 1m and 1.5m.discharges water at

outlet with a velocity of 2.5m/s. the total length of the draft tube is 6m.and 1.2m.of the

length of the draft tube is immersed in water .if the atmospheric draft tube is equal to 0.2into

velocity head at the outlet of the tube, find (a) pressure head at the inlet and (b) efficiency of

the draft tube.

[Ans. 4.27m. absolute, 76.3%]

ON KAPLAN TURBINE:

1. An axial flow hydraulic turbine has a net head of 23 m. across it, and, when running at a

speed of 150 rpm, develops 23MW. The blade tip and hub diameter are 4.75 m. and 2.0

m. respectively. If the hydraulic efficiency is93%and the overall efficiency85%, calculate

the inlet and outlet blade angle at the mean radius assuming axial flow at outlet.

[Ans. 156.20, 17.20]

2. A Kaplan turbines operating under a net head of 20 m. develop 16MW with an overall

efficiency of 80%. The diameter of the runner is 4.2 m. while the hub diameter is 2 m.

and the ‘dimensionless power specific speed’ is 3 radian. If the hydraulic efficiency is

90% calculate the inlet and exit angles of the runner blades at the tip and at the hub if the

flow leaving the runner is purely axial.

[Ans. 163.40, 14.50; 127.60, 28.60]