We have already seen the derivation of continuity
equation, Bernoulli’s
equation, momentum
equation, velocity
of sound in an isothermal process, velocity
of sound in an adiabatic process, 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 and force exerted by a jet on stationary curved plate in our recent post.
Now we will see here the derivation of expression of
force exerted by a jet on a hinged plate 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 hinged plate.
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 hinged plate
Let us consider a jet of water striking a vertical
plate at the center which is hinged at point O as displayed here in following
figure.
When jet of water will strike the plate, the plate
will swing through an angle about the hinge as displayed here in following
figure.
Let us consider the following terms from above
figure.
x = Distance of center of jet from hinge O
θ = Angle of swing about hinge
W = Weight of plate acting at C.G of the plate
Initial position of the vertical plate is displayed
above in figure by dotted lines. When water jet will strike the plate, the
plate will swing through an angle θ. After the jet strikes the plate, point A
will be now at point A’ as displayed in above figure.
Distance OA = Distance OA’ = x
Weight of plate will be acted through the point A’
after water jet strikes the plate.
There will be two forces acting on the plate as
mentioned here.
- Force due to jet of water normal to the plate, Fn
- Weight of the plate, W
Force due to jet of water normal to the plate
Fn = ρ a V2 Sin (90 – θ)
Weight of the plate, W
Once the jet strikes the plate, for equilibrium
condition we will have following equation as mentioned here.
Moment of force Fn about hinge = Moment
of weight W about hinge
Moment of force Fn about hinge = Fn
x OB = ρ a V2 Sin (90 – θ) x
OB
Moment of force Fn about hinge = ρ a V2
Cos θ x OB
Moment of force Fn about hinge = ρ a V2
Cos θ x (OA / Cos θ)
Moment of force Fn about hinge = ρ a V2
x OA
Moment of force Fn about hinge = ρ a V2
x
Now we will determine the moment of weight W about
hinge
Moment of weight W about the hinge = W x OA’ Sin θ =
W x Sin θ
Once the jet strikes the plate, for equilibrium
condition, we will have
Moment of force Fn about hinge = Moment
of weight W about hinge
ρ a V2 x = W x Sin θ
Sin θ = ρ a V2 /W
Above equation, derived here, provides the angle of
swing of plate when water jet strikes the vertical plate which is fixed at
hinge.
Do you have any suggestions? Please write in comment
box.
Reference:
Fluid mechanics, By R. K. Bansal
Image courtesy: Google
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