40
1 Working Principles
1.9 Exercises
1.9.1 The left figure shows a liquid jet impacting a plate perpendicularly in an
open space. Use a momentum balance to determine the force on the plate in the direction of the oncoming flow. Assume that the flow, after turning, perfectly follows
the plate. Assume steady flow and ignore friction. Consider gravity perpendicular
to the jet, as indicated. Use a momentum balance to determine the force on the plate
in the vertical direction. Note that the force approximately equals zero. Verify if the
result is independent of the size of the chosen control volume. Reason that the force
into the vertical direction exactly equals zero in the absence of friction on the plate.
The right figure shows a liquid jet impacting the blade of a Pelton turbine (tangential hydraulic turbine, see Chap. 9). The blade has the shape of a double spoon.
The jet leaves the blade with the indicated angle a.
Determine first the force exerted on the blade into the direction of the jet when
the blade stands still and assuming that gravity is perpendicular to the figure, i.e. not
as indicated in the figure. Determine then the force components in vertical and horizontal directions exerted on the blade assuming that gravity is as indicated in the
figure. Verify if the result is independent of the size of the chosen control volume.
Note that the force in the vertical direction is very weak. May we conclude that the
force exactly equals zero?
A: I: x
y
F mv, F
gAL,
r
=
= −
with L d h 0
≈ − ≈ .
II:
with
1.9.2 The figure is a sketch of a hydrofoil boat with speed v = 20 m/s. The mass
of the boat is 100 t. The foils that keep the vessel above the surface have a lift/
drag ratio L/D = 20. Water is sucked in by a pump and ejected through a nozzle
at 45 m/s relative to the boat. Determine the mass flow rate to be handled by the
x
x
y
F mv( 1 cos ); F mv( 1 cos ),F
gAL,
a
a
r
=
+
≈
+
= −
h
L d
2h sin .
sin
a
a
≈ +
−
1 Working Principles
1.9 Exercises
1.9.1 The left figure shows a liquid jet impacting a plate perpendicularly in an
open space. Use a momentum balance to determine the force on the plate in the direction of the oncoming flow. Assume that the flow, after turning, perfectly follows
the plate. Assume steady flow and ignore friction. Consider gravity perpendicular
to the jet, as indicated. Use a momentum balance to determine the force on the plate
in the vertical direction. Note that the force approximately equals zero. Verify if the
result is independent of the size of the chosen control volume. Reason that the force
into the vertical direction exactly equals zero in the absence of friction on the plate.
The right figure shows a liquid jet impacting the blade of a Pelton turbine (tangential hydraulic turbine, see Chap. 9). The blade has the shape of a double spoon.
The jet leaves the blade with the indicated angle a.
Determine first the force exerted on the blade into the direction of the jet when
the blade stands still and assuming that gravity is perpendicular to the figure, i.e. not
as indicated in the figure. Determine then the force components in vertical and horizontal directions exerted on the blade assuming that gravity is as indicated in the
figure. Verify if the result is independent of the size of the chosen control volume.
Note that the force in the vertical direction is very weak. May we conclude that the
force exactly equals zero?
A: I: x
y
F mv, F
gAL,
r
=
= −
with L d h 0
≈ − ≈ .
II:
with
1.9.2 The figure is a sketch of a hydrofoil boat with speed v = 20 m/s. The mass
of the boat is 100 t. The foils that keep the vessel above the surface have a lift/
drag ratio L/D = 20. Water is sucked in by a pump and ejected through a nozzle
at 45 m/s relative to the boat. Determine the mass flow rate to be handled by the
x
x
y
F mv( 1 cos ); F mv( 1 cos ),F
gAL,
a
a
r
=
+
≈
+
= −
h
L d
2h sin .
sin
a
a
≈ +
−