FORCE EXERTED ON A SERIES OF RADIAL CURVED VANES

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, force exerted by a jet on a curved plate and force exerted by a jet ofwater on a series of vanes in our recent posts.

Now we will see here the derivation of expression of force exerted by a jet of water on a series of radial curved 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 radial curved 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 radial curved 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.

Let us see here the condition of impact of jet on a series of radial curved vanes mounted on a wheel as displayed here in following figure.

For a radial curved vane, the radius of the vane at inlet and outlet will be different and therefore the tangential velocities of the radial vane at inlet and outlet will be different.

Jet of water will strike the vanes and the wheel will start rotating at a constant angular speed. Let us consider the following terms as mentioned here.

R₁ = Radius of wheel at the inlet of vane

R₂ = Radius of wheel at the outlet of vane

ω = Angular speed of the wheel

Velocity triangles at the inlet and outlet are drawn here in above figure.

Efficiency of the radial curved vane

Work done per second on the wheel will be considered as the output of the system and initial kinetic energy per second of the jet will be taken as input of the system.

We can conclude here the efficiency of the radial curved vane as mentioned here.

If there is no loss of energy when water is flowing over the vanes, the work done on the wheel per second will be equal to the change in kinetic energy of the jet per second.

Work done per second on the wheel = Change in kinetic energy of the jet per second

From the above expression, we can say that for a given initial velocity of the jet i.e. V₁, the efficiency will be maximum when V₂ will be minimum. But same time we can also conclude that V₂ could not be zero, as in that condition incoming jet will not move out of the vane.

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

We will see another topic i.e. Jet propulsion, in the subject of fluid mechanics, with the help of our next post.