We have already seen the derivation and fundamentals
of stagnation
properties i.e. stagnation pressure, stagnation temperature and stagnation
density for compressible fluid flow in our previous posts.
We have also discussed 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 and force exerted by a jet on a hinged plate and in our recent post.
Now we will see here the derivation of expression of
force exerted by a jet on a curved plate when the plate is moving in the
direction of jet 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 on
a curved plate when the plate is moving in the direction of jet.
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 on a curved plate when the plate is moving in the direction of jet
Let us consider a jet of water striking a curved
plate at the center of the plate which is moving with a uniform velocity in the
direction of jet as displayed here in following figure.
Let us consider the following terms from above
figure
V = Absolute velocity of jet
a = Area of jet
u = Velocity of the plate in the direction of jet
Relative velocity of water jet or velocity with
which the water jet strikes the curved plate = (V-u)
Let us consider that plate is smooth and loss of
energy due to the impact of jet is zero. We will have the similar velocity with
which the jet will be leaving the curved plate i.e. (V-u)
Velocity will be resolved into two components. One
component will be in the direction of jet and second component will be in the
direction perpendicular to the jet.
Component of the velocity in the direction of jet = -
(V- u) Cos θ
We have taken negative sign as at the outlet, the
component will be in the opposite direction of jet.
Component of the velocity in the direction perpendicular
to the jet = (V- u) Sin θ
Mass of the water striking the plate = ρ a (V-u)
Force exerted by the jet of water on the curved
plate in the direction of jet
Work done by the jet per second on the plate will be
given by following equation
Above equation, derived above, force exerted by the
jet of water on the curved plate in the direction of jet and works done by the
jet per second on the plate when water jet strikes the curved plate when the plate is moving in the direction of
jet.
Do you have any suggestions? Please write in comment
box.
Reference:
Fluid mechanics, By R. K. Bansal
Image courtesy: Google
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