19 Dec

# Introduction

One way of producing mechanical work from a fluid under pressure is to use the pressure to accelerate the fluid to a high velocity in a jet. The jet is directed onto the vanes of a turbine wheel that is rotated by the force generated on the vanes due to the momentum change or impulse which takes place as the jet strikes the vanes. Water turbines working on this impulse principle have been constructed with outputs of the order of 100 000 kW and with efficiencies greater than 90%.

# Object

1. To measure the force produced by a water jet when it strikes two types of vane: a flat plate and a hemispherical cup.

1. To compare the results with the theoretical values calculated from the momentum flux in the jet.

# Theory

When a jet of fluid strikes a vane and is deflected, a force is generated due to the change in momentum of the fluid. The force is given by the momentum equation which states that the force on the fluid is equal to the rate of momentum change of the fluid. This can be expressed simply as the difference between the initial and final momentum flow rates (or momentum fluxes).

For the general case shown in Figure 1 the momentum flux in the jet is where is the mass flow rate and is the jet velocity just upstream of the vane. After being deflected through an angle b, the momentum flux is  in the x direction. The force on the fluid is therefore in the x direction. The force F in the x direction on the vane is therefore

.                                          (1)

 u1

Figure 1 Flow of a jet over a curved vane.

In the case of a flat plate (Figure 2a), b = 90°, so cosb = 0 and equation (1) reduces to

(2)

irrespective of the value of u1.

For a hemispherical cup (Figure 2b), b = 180°, so cosb = -1 and equation (1) reduces to

.

Furthermore, if there is negligible reduction in speed so that u1 = u0 then the maximum possible force on the hemispherical cup is

(3)

(a) Flat Plate                                        (b) Hemispherical Cup

Figure 2 Flow of a jet over a flat plate and hemispherical cup

In this experiment it is not possible to measure directly the velocity just upstream of the vane. However, the velocity u at the exit of the nozzle can be determined. The velocity u0 is less than u because of the deceleration due to gravity and can be calculated from the equation

(4)

where s is the distance between the nozzle exit and the surface of the vane.

# Procedure

1. Fit the flat plate to the apparatus. If the cup is fitted, remove it by undoing the retaining screw and lifting it out with the cover plate. Take care not to drop the cup in the plastic cylinder.
2. Fit the cover plate over the stem of the flat plate and hold it in position below the beam. Screw in the retaining screw and tighten it.
3. Set the weigh-beam to its datum position. First set the jockey weight on the beam so that the datum groove is at zero on the scale, Figure 3. Turn the adjusting nut, above the spring, until the grooves on the tally are in line with the top plate as shown in Figure 4. This indicates the datum position to which the beam must be returned during the experiment, to measure the force produced by the jet.
4. Switch on the bench pump and open the bench supply valve to admit water to the apparatus. Check that the drainpipe is over the hole leading to the weighing tank.
5. Fully open the supply valve and restore the balance position by sliding the jockey weight along the beam until the grooves on the tally are in line with the top plate. Record the reading on the scale corresponding to the groove on the jockey weight.
6. Measure the mass flow rate.  As the water fills the weighting tank, the hopper balance bar will start to rise and reach a balance point. At this time, start the stopwatch and apply a 5kg weight to the hanger. Record the time it takes from applying the weight until the hopper balance bar is horizontal again (reaches the stop). (From figure 4 you will see that a 5 kg mass will require 15kg of water to enter the weighting tank).
7. Close the water supply valve, remove the weight and drain the weighing tank.
8. Move the jockey weight inwards by 10 (when using the flat plate) or 15 mm (when using the hemispherical cup). Open the water supply valve and adjust the flow rate until the beam is approximately level (when the grooves on the tally are in line with the top plate). If needed the beam can then be further adjusted to a level position by moving the jockey weight. Record the reading on the scale corresponding to the groove on the jockey weight. Repeat steps 6 and 7.
9. Repeat step 8 until you have about 6 sets of readings over the range flow. For the last set, the jockey should be set at about 10 mm from the zero position.
10. Switch off the bench pump and fit the hemispherical cup to the apparatus using the method in steps 1 and 2.
11. Repeat steps 3-9 using the hemispherical cup.
12. Switch of the bench pump and record the mass m of the jockey weight, the diameter d of the nozzle, and the distance s of the vanes from the outlet of the nozzle.

Figure 3 The jet impact apparatus showing the major components.

 Hanger with 5kg Mass
 Stop
 Hopper Balance Bar
 Flat Plate Target
 Hemispherical Cup Target
 Spring with Adjustable Tension for Levelling Balance Beam w/ No Flow
 Jockey Balance Bar
 0
 Water Reservoir
 Weighing Tank (Hopper)
 Jockey Mass
 Valve
 P

Figure 4 Schematic diagram of the impact water jet machine

# Data

Record the following data:

Mass of jockey                                                                           =                 kg

Distance from centre-line of vane to pivot of lever             =                 m

Diameter of nozzle (d)                                                              =               mm

Height of vane above nozzle outlet (s)                                 =               mm

# Calculations and results

1. Taking moments about the pivot of the lever and substituting known values into the equation, it can be shown that the force F is given by:

F = 4gy    newtons

where y is the displacement of the jockey weight in metres. Calculate the force for each condition.

1. Calculate the values for the mass flow rate (in kg/s) and for each of these, determine the velocity u of the jet.

(r for water = 103 kg/m3)

where  a is the nozzle area in m2.

1. Using equation (4) determine the velocities just upstream of the jet

m/s

1. Calculate the momentum flux in the jet for each case.

1. Tabulate all results and plot F against for the two vane shapes.

1. The theoretical values of F are

Flat plate                    :

Hemispherical cup  :

Compare these values with the measured values on your graphs.

Some discussion points

1. How well do your results compare with theory?

1. Why is the force on the hemispherical cup somewhat less than twice that on the flat plate?

1. What accuracy have you achieved in measuring the force on each of the vanes?

1. If the lines on your graph do not pass through the origin, what reason might there be for this?

References

Webber, N B                                                         “Fluid Mechanics for Civil

Engineers”, Chapman and Hall

Ltd, London

Douglas, J F, Gasiorek J M and Swaffield J A   “Fluid Mechanics”, Pitman Ltd,

London

Massey B S                                                          “Mechanics of Fluids”, Van

Nostrand Reinhold Ltd, London

# Introduction

One way of producing mechanical work from a fluid under pressure is to use the pressure to accelerate the fluid to a high velocity in a jet. The jet is directed onto the vanes of a turbine wheel that is rotated by the force generated on the vanes due to the momentum change or impulse which takes place as the jet strikes the vanes. Water turbines working on this impulse principle have been constructed with outputs of the order of 100 000 kW and with efficiencies greater than 90%.

# Object

1. To measure the force produced by a water jet when it strikes two types of vane: a flat plate and a hemispherical cup.

1. To compare the results with the theoretical values calculated from the momentum flux in the jet.

# Theory

When a jet of fluid strikes a vane and is deflected, a force is generated due to the change in momentum of the fluid. The force is given by the momentum equation which states that the force on the fluid is equal to the rate of momentum change of the fluid. This can be expressed simply as the difference between the initial and final momentum flow rates (or momentum fluxes).

For the general case shown in Figure 1 the momentum flux in the jet is where is the mass flow rate and is the jet velocity just upstream of the vane. After being deflected through an angle b, the momentum flux is  in the x direction. The force on the fluid is therefore in the x direction. The force F in the x direction on the vane is therefore

.                                          (1)

 u1

Figure 1 Flow of a jet over a curved vane.

In the case of a flat plate (Figure 2a), b = 90°, so cosb = 0 and equation (1) reduces to

(2)

irrespective of the value of u1.

For a hemispherical cup (Figure 2b), b = 180°, so cosb = -1 and equation (1) reduces to

.

Furthermore, if there is negligible reduction in speed so that u1 = u0 then the maximum possible force on the hemispherical cup is

(3)

(a) Flat Plate                                        (b) Hemispherical Cup

Figure 2 Flow of a jet over a flat plate and hemispherical cup

In this experiment it is not possible to measure directly the velocity just upstream of the vane. However, the velocity u at the exit of the nozzle can be determined. The velocity u0 is less than u because of the deceleration due to gravity and can be calculated from the equation

(4)

where s is the distance between the nozzle exit and the surface of the vane.

# Procedure

1. Fit the flat plate to the apparatus. If the cup is fitted, remove it by undoing the retaining screw and lifting it out with the cover plate. Take care not to drop the cup in the plastic cylinder.
2. Fit the cover plate over the stem of the flat plate and hold it in position below the beam. Screw in the retaining screw and tighten it.
3. Set the weigh-beam to its datum position. First set the jockey weight on the beam so that the datum groove is at zero on the scale, Figure 3. Turn the adjusting nut, above the spring, until the grooves on the tally are in line with the top plate as shown in Figure 4. This indicates the datum position to which the beam must be returned during the experiment, to measure the force produced by the jet.
4. Switch on the bench pump and open the bench supply valve to admit water to the apparatus. Check that the drainpipe is over the hole leading to the weighing tank.
5. Fully open the supply valve and restore the balance position by sliding the jockey weight along the beam until the grooves on the tally are in line with the top plate. Record the reading on the scale corresponding to the groove on the jockey weight.
6. Measure the mass flow rate.  As the water fills the weighting tank, the hopper balance bar will start to rise and reach a balance point. At this time, start the stopwatch and apply a 5kg weight to the hanger. Record the time it takes from applying the weight until the hopper balance bar is horizontal again (reaches the stop). (From figure 4 you will see that a 5 kg mass will require 15kg of water to enter the weighting tank).
7. Close the water supply valve, remove the weight and drain the weighing tank.
8. Move the jockey weight inwards by 10 (when using the flat plate) or 15 mm (when using the hemispherical cup). Open the water supply valve and adjust the flow rate until the beam is approximately level (when the grooves on the tally are in line with the top plate). If needed the beam can then be further adjusted to a level position by moving the jockey weight. Record the reading on the scale corresponding to the groove on the jockey weight. Repeat steps 6 and 7.
9. Repeat step 8 until you have about 6 sets of readings over the range flow. For the last set, the jockey should be set at about 10 mm from the zero position.
10. Switch off the bench pump and fit the hemispherical cup to the apparatus using the method in steps 1 and 2.
11. Repeat steps 3-9 using the hemispherical cup.
12. Switch of the bench pump and record the mass m of the jockey weight, the diameter d of the nozzle, and the distance s of the vanes from the outlet of the nozzle.

Figure 3 The jet impact apparatus showing the major components.

 Hanger with 5kg Mass
 Stop
 Hopper Balance Bar
 Flat Plate Target
 Hemispherical Cup Target
 Spring with Adjustable Tension for Levelling Balance Beam w/ No Flow
 Jockey Balance Bar
 0
 Water Reservoir
 Weighing Tank (Hopper)
 Jockey Mass
 Valve
 P

Figure 4 Schematic diagram of the impact water jet machine

# Data

Record the following data:

Mass of jockey                                                                           =                 kg

Distance from centre-line of vane to pivot of lever             =                 m

Diameter of nozzle (d)                                                              =               mm

Height of vane above nozzle outlet (s)                                 =               mm

# Calculations and results

1. Taking moments about the pivot of the lever and substituting known values into the equation, it can be shown that the force F is given by:

F = 4gy    newtons

where y is the displacement of the jockey weight in metres. Calculate the force for each condition.

1. Calculate the values for the mass flow rate (in kg/s) and for each of these, determine the velocity u of the jet.

(r for water = 103 kg/m3)

where  a is the nozzle area in m2.

1. Using equation (4) determine the velocities just upstream of the jet

m/s

1. Calculate the momentum flux in the jet for each case.

1. Tabulate all results and plot F against for the two vane shapes.

1. The theoretical values of F are

Flat plate                    :

Hemispherical cup  :

Compare these values with the measured values on your graphs.

Some discussion points

1. How well do your results compare with theory?

1. Why is the force on the hemispherical cup somewhat less than twice that on the flat plate?

1. What accuracy have you achieved in measuring the force on each of the vanes?

1. If the lines on your graph do not pass through the origin, what reason might there be for this?

References

Webber, N B                                                         “Fluid Mechanics for Civil

Engineers”, Chapman and Hall

Ltd, London

Douglas, J F, Gasiorek J M and Swaffield J A   “Fluid Mechanics”, Pitman Ltd,

London

Massey B S                                                          “Mechanics of Fluids”, Van

Nostrand Reinhold Ltd, London