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 ρaV3
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
Image courtesy: Google
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