When a jet of water strikes on target of various shapes, a force is generated with the consistent of conservation of linear momentum. The objective of this experiment is to find out the validity of theoretical value of impact of jet when hits target with various shape. 1200 cone shape is chosen as target to be observed in this experiment. Initially the balance indicator is reset when there is no load on the balance. A mass of 10g is placed on the pan to press the balance down so that it moves away from its balance position. Next, a jet of water is used to strike the 1200 inverted cone on from the opposite of mass load. Flow rate of water is adjusted so that the impact of water jet strikes the balance indicator to its original position. The flow rate of water is determined by calculating the time taken to collect 5L of water. The time taken to collect 5L of water for same mass is determined twice to increase the accuracy. The experiment is repeated by using successive increment of mass load. The data obtained is tabulated and a suitable graph is plotted for further analysis. The experimental value is compared with theoretical value together with errors in the experiment. Finally, conclusion is made base on the validity of theoretical value compared with experimental value.

2 Introduction

One of the applications of jet impact is used to generate electricity. Impact of jet is used to rotate the turbine namely Pelton wheel in the generator. The water jet is applying force tangential to the wheel. The tangential forces of water jet generate moment or torque on the wheel to its maximum value and therefore increase the mechanical energy of Pelton wheel. The rotational energy of wheel is then converted into electrical energy.

Figure 1: Pelton wheel

(Image source: http://www.make-energy.info/wp-content/uploads/2009/01/hield1.jpg)

Besides, another application of water jet is water jet cutter. Water jet cutter is capable to slice metal or other material by using the high pressure and high velocity of water jet together with abrasive substance. The process of water jet cutting is similar with water erosion found naturally but occurs with highly accelerated and concentrated.

There are several advantages on water jet cutting machine:

1. No heat generated. This is especially useful when cutting material which sensitive towards heat where excessive heat will change the properties of material.

2. Water jet cutting does not generate any dust or particles that are harmful when inhaled.

3. Water jet cutter can easily automate for production use.

4. Water jet cutting can produce a smooth edge and remove other machining operation such as finish sanding and grinding.

5. Weight of water jet machine is much less than laser cutter. When installed on robot, the problem of accelerating and decelerating robot head can be reduce and less energy consumed.

6. Kerf width in water jet cutting is very small, and very little material is wasted. (http://www.mfg.mtu.edu)

Figure 2: Abrasive water jet cutter

(Image source: http://en.wikipedia.org/wiki/Water_jet_cutter)

Before the derivation of equation, there are some assumptions made:

1. The contact of water particles and target surface are frictionless.

2. When the molecules of water strike on target surface, the surface will maintain a constant angle with the jet of water that is the target surface will remain in place.

3. The collision of water molecules and target surface are perfectly elastic. The kinetic energy and linear momentum of water jet are therefore conserved.

4. The magnitude of force of water jet remain unchanged as it emerge out from nozzle and strikes on the target surface. The plane is placed as close as possible to the nozzle so that the effect of gravitational acceleration on water jet can be neglected.

According to newton second law:

1. The resultant force of net force act on a system is equal to the rate of change of momentum of the system.

2. The object is accelerating with the same direction as the resultant force.

Figure 3: Set up of apparatus

(Image source: http://www.mae.cemr.wvu.edu/laboratories/images/fluidjet.jpg)

Mathematical derivation:

F

F= m

F = (v2 – v1)

Where

F= resultant force exerted on stream of water (N)

m= mass of stream of water (kg)

v1=incident velocity of the stream of water (ms-1)

v2=reflected velocity of the stream of water (ms-1)

t = time (s)

Since the collision of water molecules with the wall of target surface is assumed to be elastic, the magnitude of v1 and v2 are therefore same, that is |v1|=|v2|=v

F= (-v cos α – v)

According to newton’s third law of motion, every action will have reaction that opposes the action with same magnitude. Hence, the force act on the target plane, Fy will have the same magnitude of force as the force act on the stream of water but in opposite direction. For the newton’s third law of motion to be applicable, there must be two objects interact with each other. In the experiment, the two objects interact with each other are water molecules and the surface of target plane.

Therefore,

Fy= -F

Fy= (v cos α + v)

Where v cos α is the vertical component of the reflected stream of water particles.

Fy= (v cos α + v)

The rate of change of mass can be equate with the product of density of water and rate of flow water as the dimension of density is mass per unit volume while the dimension of rate of water flow is volume per unit time. When multiplying both values we can get homogeneous dimension with rate of change of mass that is mass per unit time.

¡ = 180° – ±

Fy = ²Q [v cos (180° – ±) + v]

Fy = ²Q (v – v cos ±)

Where,

Fy = Vertical component of the force exerted on the target plane (N)

² = density of water (Kg m-3)

Q = Rate of flow of the water (m3s-1)

± = angle of slope of target plane (°)

¡ = reflected angle of the stream of water (°)

The velocity of stream water can be equate to the volume flow rate per unit cross sectional area perpendicular to the direction of stream line, v=

Where A = cross sectional area of the nozzle where the water flow out.

Therefore,

Fy =

And =cos120

=

Fy =

œ Fy =

However, Fy = Mg, when the weight has been pushed back to its equilibrium position.

Where, M is the mass of load on the pan .Hence, M =

3 Objective

To investigate the validity of theoretical expressions of force exerted on targets of various shapes by a jet.

4 Results and calculations

For 1200 cone target:

Volume of water, V= 5 litres

= 5×10-3 m3

Diameter of nozzle, d= 8mm

Cross sectional area of nozzle =

=

=5.027×10-5m2

Table 1

Load mass

,m(g)

Load weight,

W(N)

Time, t(s)

Flow rate,

Q(x10-4m3s-1)

Q2,

(x10-8m6s-2)

Jet force,

Fy

(N)

1

2

average

10

0.0981

66.04

64.66

65.35

0.765

0.585

0.0981

20

0.1962

60.38

59.22

59.80

0.836

0.699

0.1962

50

0.4905

46.38

44.18

45.28

1.104

1.219

0.4905

100

0.9810

31.90

32.50

32.20

1.553

2.411

0.9810

130

1.2753

28.31

28.28

28.30

1.767

3.123

1.2753

150

1.4715

24.03

22.53

23.28

2.148

4.613

1.4715

180

1.7658

22.00

20.24

21.12

2.367

5.605

1.7658

From the graph, gradient, mexperimental = 3.38x 107 Nm-6s2

Theoretical value of gradient is given by,

mtheoretical=

= 2.98x107Nm-6s2

Hence, the percentage of error for the gradient of experimental and theoretical value:

5 Discussions

From the result obtained, the experimental data is approximately same with the theoretical data, which is about 13.42% of percentage error. Since the percentage error is less than 15%, the theoretical value of force exerted by jet on a 1200 curved target surface is accepted.

However, there is still exist many discrepancy between experimental value and theoretical value. These discrepancy is mainly due to the errors and assumptions when conducting the experiment. Firstly, the effect of gravity on the water jet is neglected. In fact, the velocity of water jet will decrease as it leaves the nozzle and before it collide with the target surface. The decrease in velocity due to the acceleration due to gravity will cause more mass is needed to produce effect of same flow rate.

The another assumption made is the contact of water molecules and target surface are assume to be frictionless. In fact, movement of water along the targeted surface is not totally frictionless, energy is loss in the form of heat.

Morever, the assumption of elastic collision between water molecules and wall aslo cause the discrepency of experiment. The collision between water molecule and target surface is not completely elastic and therefore some of the energy transform into heat energy due to friction.

Apart from that, the actual force exerted by water jet to targeted surface less than the theoretical value. This is due to some of the energy is converted into pontential energy when travel from nozzle to wall.

In addition, parallax error is associate with observer when observing the return of balance indicator toward original position. Because of the limitation of human eye, the position of balance point may not be so accurate.

6 CONCLUSION

The theoretical expressions for the force exerted by a jet on target is considered valid for 120° cone target since the percentage of error is relatively small which is 13.42%.