We have seen the fundamentals of impact of jets, force exerted by a jet on vertical flat plate,  force exerted by a jet on stationary inclined flat plate, force exerted by a jet on stationary curved plate, force exerted by a jet on a hinged plate and  force exerted by a jet on a curved plate when the plate is moving in the direction of jet in our recent post.

Now we will see here the derivation of expression of force exerted by a jet of water on a series of vanes with the help of this post. Let us first brief here the basic concept of impact of jets and after that we will derive the expression of force exerted by a jet of water on a series of vanes.

### Impact of jets

Let us consider that we have one pipe through which liquid is flowing under pressure. Let us assume that a nozzle is fitted at outlet of pipe. Liquid which will come through the outlet of nozzle will be in the form of jet.

If a plate, which may be moving or fixed, is placed in the path of jet, there will be one force which will be exerted by the jet over the surface of plate. The force which will be exerted by the jet over the surface of plate, which might be moving or fixed, will be termed as impact of jet.

### Force exerted by a jet of water on a series of vanes

If we see practically, force exerted by a jet of water on a single moving plate will not be feasible. Therefore, we will see the practical case where large number of plates will be mounted on the circumference of a wheel at a fixed distance apart as displayed here in following figure.

Jet will strike a plate and due to the force exerted by the jet on plate, wheel will be started to move and therefore second plate mounted on the circumference of wheel will be appeared before the jet and jet will again exert the force to the second plate.

Therefore, each plate will be appeared successively before the jet and jet will strike each plate or jet will exert force to each plate. Therefore, wheel will be rotated with a constant speed.

Let us consider the following terms as mentioned here
V = Velocity of jet
d = Diameter of jet
a = Cross-sectional area of jet = (π/4) x d2
u = Velocity of vane

Mass of water striking the series of plate per second = ρaV

Jet strikes the plate with a velocity = V-u

After striking, jet will move tangential to the plate and therefore velocity component in the direction of motion of plate will be zero.

Force exerted by the jet in the direction of motion of plate
FX = Mass striking the series of plate per second x [Initial velocity – final velocity]
FX = ρaV [(V-u)-0] = ρaV (V-u)

Work done by the jet on the series of plate per second = Force x Distance per second in the direction of force

Work done by the jet on the series of plate per second = FX x u = ρaV (V-u) x u

Kinetic energy of the jet per second = (1/2) x mV2
Kinetic energy of the jet per second = (1/2) x ρaV V2
Kinetic energy of the jet per second = (1/2) x ρaV

Efficiency = Work done per second / Kinetic energy per second
Efficiency = ρaV (V-u) x u / (1/2) x ρaV3
Efficiency = 2 u (V-u)/V2

Maximum efficiency will be 50 % and it will be when u = V/2

Do you have any suggestions? Please write in comment box.

We will see another topic i.e. Force exerted on a series of radial curved vanes in the subject of fluid mechanics with the help of our next post.

Reference:
Fluid mechanics, By R. K. Bansal
1. 2. 