The curved plate moving of jet
Force on the curved plate when the plate is moving in the direction of the jet
Let a jet of water strikes a curved plate at the centre of the plate which is moving with a uniform velocity in the direction of the jet as shown in Fig.
 |
| Image Source ~ Crafted With ©Ishwaranand - 2020 ~ Image by ©Ishwaranand |
Fig. Jet striking a curved moving plate
Let
v = Absolute velocity of jet,
a = Area of a jet,
u = Velocity of the plate in the direction of the jet.
The relative velocity of the jet of water or the velocity with which jet strikes the curved plate
= (v - u)
If the plate is smooth and the loss of energy due to impact of a jet is zero, then the velocity with which the jet will be leaving the curved vane
= (v - u)
This velocity can be resolved into two components, one in the direction of the jet and other perpendiculars to the direction of the jet.
(-ve sign is taken as at the outlet, the component is in the opposite direction of the jet).
Mass of the water striking the plate
= ρa x velocity with which jet strikes the plate
= ρa (v –u)
Force exerted by a jet of water on the curved plate in the direction of the jet, Fx
Fx = Mass striking per second x [Initial velocity with which jet strikes the plate in the direction of jet – Final velocity]
= ρa(v – u) [(v – u) – (– (v – u) cosθ)]
= ρa(v – u) [(v – u) + (v – u) cosθ]
= ρa(v – u)^2 [ 1 + cosθ]
Work is done by the jet on the plate per second
= Fx x Distance travelled per second in the direction of x
= Fx x u
= ρa(v – u)^2 [ 1 + cosθ] x u
= ρa(v – u)^2 u [ 1 + cosθ]
Ex. A jet of water of diameter 7.5 cm strikes a curved plate at its centre with a velocity of 20 m/s. This curved plate is moving by a velocity of 8 m/s in the direction from the jet. Each jet is deflected through an angle of 165°. Considering the plate smooth,
Find
- Force exerted at the plate into the direction from the jet,
- Power of the jet, and
- The efficiency of the jet.
Given
The diameter of the jet,
d = 7.5 cm
= 0.075 m
Area,
a = (π/4) (0.075)^2
= 0.004417
The velocity of the jet,
v = 20 m/s
The velocity of the plate,
u = 8 m/s
The angle of deflection of the jet
= 165°
Solution
Angle made by the relative velocity at the outlet of the plate,
θ = 180° - 165°
= 15°
Force exerted by the jet on the plate in the direction of the jet is given by,
Fx = ρa (v - u)^2 (1 + cosθ)
= 1000 x 0.004417 x (20 - 8)^2 [1 + cos15°]
= 1250.38 N
Work done to the jet on this plate is /sec
= Fx x u
= 1250.38 x 8
= 10003.04 N m/s
Power of the jet
= 10003.04/1000
= 10 kW
= Output / Input
= (Work done by jet/sec) / ( Kinetic energy of jet/sec )
= (1250.38 x 8) / 1/2(pav^3)
= (1250.38 X 8) / 1/2(1000 x 0.004417 x 20^3)
= 56.4%
