Impact of Jet on a Plate or Vane: Notes and Viva Questions.
Introduction to Jet:
A jet of water issuing from a nozzle has a velocity and hence it possesses a kinetic energy. If this jet strikes a plate then it is said to have an impact on the plate. The jet will exert a force on the plate which it strikes. This force is called a dynamic force exerted by the jet. This force is due to the change in the momentum of the jet as a consequence of the impact. This force is equal to the rate of change of momentum i.e., the force is equal to (mass striking the plate per second) x (change in velocity).
We will consider some particular cases of impact of a jet on a plate or vane.
(a) Direct Impact of a Jet on a Stationary Flat Plate:
ADVERTISEMENTS:
Consider a jet of water impinging normally on a flat plate at rest.
Let,
a = Cross-sectional area of the jet in metre2.
ADVERTISEMENTS:
V = Velocity of the jet in metres per second.
M = Mass of water striking the plate per second.
∴ M = ρaV kg/sec
where ρ = density of water in kg/cum
ADVERTISEMENTS:
Force exerted by the jet on the plate-
P = Change of momentum per second
= (Mass striking the plate per second) x (Change in velocity)
= M (V – 0) = MV = ρaV.V.
ADVERTISEMENTS:
∴ P = ρaV2 Newton
(b) Oblique Impact of a Jet on a Stationary Flat Plate:
Let,
θ = angle between the jet and the plate.
ADVERTISEMENTS:
Velocity component normal to the plate before impact = V sin θ
Velocity component normal to the plate after impact = 0.
∴ Force exerted by the jet normal to the plate
P = (Mass striking the plate per second) x (Change in velocity normal to the plate)
P = M (V sin θ – 0) = ρaV.V sin θ
∴ P = ρaV2 sin θ Newton
Note – In the two cases discussed above, the work done by the jet on the plate is zero since the point of application of the force does not move.
(c) Direct Impact of a Jet on a Moving Plate:
Let,
V = Velocity of the jet
v = Velocity of the plate.
Velocity of the jet relative to the plate = (V – v)
We may consider as though the plate is at rest and that the jet is moving with a velocity (V –v) relative to the plate.
∴ Force exerted by the jet on the plate
= P = ρa (V – v)2 Newton
In this case, since the point of application of the force moves, work is done by the jet.
Work done by the jet on the plate per second
= Pv = ρa (V – v)2 v Nm/s or Joule/sec
(d) Oblique Impact of a Jet on a Moving Vane:
Let the velocity of the jet and the vane be V and v in the same direction. Let the angle between the jet and the plate be θ. In this case mass of liquid striking the plate per second
= ρa (V – v)
Relative velocity normal to the plate before impact
= (V – v) sin θ
Relative velocity normal to the plate after impact = 0
∴ Force exerted by the jet normal to the plate
P = ρa (V – v) [(V – v) sin θ – 0]
∴ P = ρa (V – v)2 sin θ Newton
This force P acting normal to the plate can be resolved into components Px and Py in the direction of motion of plate and perpendicular to the direction of motion of the plate.
∴ Px = ρa (V – v)2 sin2 θ and
and Py = ρa (V – v)2 sin θ cos θ
∴ Work done by jet per second
= Pxv = ρa (V – v)2 v sin2 θ
Note – The cases c and d discussed above do not arise in practice since this case needs a jet which has to follow the plate in continuity as the plate goes on moving.
(e) Direct Impact of a Jet on a Series of Flat Vanes Mounted on the Periphery of a Large Wheel:
In this case, a number of flat plates are radially mounted over a wheel. The wheel is supported over a shaft, with a suitable bearing to afford easy rotation of the wheel. The jet moving at a velocity strikes the plate in succession causing the wheel to rotate.
Velocity of the jet before striking the wheel = V
Velocity of the jet after striking the wheel
= v = Velocity of the plates at the impact point.
The wheel with plates (vanes) provided as in this case and subjected to impact by a jet is called a water wheel. When the jet strikes the wheel at the bottom as in this case, the device is called an undershot water wheel. When the jet strikes the wheel at the top, the device is called an overshot water wheel.
Pressure on Fixed Curved Vane:
Fig. 18.15 shows a fixed curved vane deflecting a jet of water.
Let ab be the normal to the centre of the curved vane.
Let the jet strike the vane at an angle α with the line ab.
Let the jet leave the vane at an angle β with the line ab. The total angle by which the jet is deflected
= 180° – (α + β)
Assuming that there is no frictional resistance, the velocity of the jet has the same magnitude at inlet and at outlet.
But the direction of the velocity has changed from the jet’s position at inlet to its position at outlet.
Force exerted at right angles to ab.
= Mass flowing per second x change in velocity at right angles to ab.
W/g (V sin α – V sin β)
Jet Striking a Symmetrical Curved Vane at Rest, at the Centre of the Vane:
In this case the jet after striking the vane at the centre gets divided into two identical jets leaving the vane at the outer tips, with the velocity having the same magnitude as that of the striking jet if no resistances are offered by the vane. See Fig.18.19.
Force exerted by the jet = F = mass flowing per second x change in velocity in the direction of the striking jet
Pressure on a Moving Curved Vane:
Consider a curved vane moving with a velocity v under the action of the dynamic thrust exerted by a jet of water gliding over it.
Let V be the velocity of the jet at inlet.
Let Vr be the relative velocity of the jet with respect to the vane.
The shape of the vane and its position with respect to the jet are so designed that the jet should just glide over it, so that there will be no loss of energy due to shock. For this condition the direction of the relative velocity Vr at inlet should be along the tangent to the vane at inlet.
The relative velocity at inlet is the vectorial difference between the velocity of the jet and the velocity of the vane at inlet.
Flow over a Radial Vane:
Fig. 18.21 shows one of the series of radially provided blades mounted on the rim of a wheel.
Let,
w = Angular velocity of the wheel
r = Radius of the wheel at inlet
r1 = Radius of the wheel at outlet
v = Tangential velocity of the vane at inlet
v1 = Tangential velocity of the vane at outlet.
Consider velocity components in the direction of motion of the wheel as positive.
Consider 1 N of flow of water.
Tangential momentum of water at inlet
Jet Propulsion:
This is a principle used to utilise the reaction of a jet to propel a vessel.
Consider the tank shown in Fig. 18.44 provided with an orifice of area a. Let h be the head of water in the tank above the orifice. When the orifice is opened a jet of water is discharged with a velocity given by –
A ship may be driven through water by the propulsive force exerted by a jet of water discharging from the back or stern of the ship. The ship is provided with a pump which draws water from the region surrounding the ship and discharges it through an outlet at the stern of the ship.
The following two cases arise:
Case 1- When the inlet orifices are normal to the direction of motion of the ship (i.e., inlet orifices are provided amid ship.)
Case 2- For the condition of maximum efficiency of propulsion,
