6.1
The figure below shows a smooth curved vane attached to a rigid foundation. The jet of water, rectangular in section, 75mm wide and 25mm thick, strike the vane with a velocity of 25m/s. Calculate the vertical and horizontal components of the force exerted on the vane and indicate in which direction these components act.
[Horizontal 233.4 N acting from right to left. Vertical 1324.6 N acting downwards]
From the question:
Calculate the total force using the momentum equation:
Body force and pressure force are 0.
So force on vane:
6.2
A 600mm diameter pipeline carries water under a head of 30m with a
velocity of 3m/s. This water main is fitted with a horizontal bend which
turns the axis of the pipeline through 75 (i.e. the internal angle at the
bend is 105). Calculate the resultant force on the bend and its angle to
the horizontal.
[104.044 kN, 52 29']
From the question:
Calculate total force.
Calculate the pressure force
p1 = p2 = p = hrg = 3010009.81 = 294.3 kN/m2
There is no body force in the x or y directions.
These forces act on the fluid
The resultant force on the fluid is
6.3
A horizontal jet of water 2103 mm2 cross-section and
flowing at a velocity of 15 m/s hits a flat plate at 60 to the axis (of
the jet) and to the horizontal. The jet is such that there is no side
spread. If the plate is stationary, calculate a) the force exerted on the
plate in the direction of the jet and b) the ratio between the quantity of
fluid that is deflected upwards and that downwards. (Assume that there is
no friction and therefore no shear force.)
[338N, 3:1]
From the question a2 = a3 =2x10-3 m2 u = 15 m/s
Apply Bernoulli,
Change in height is negligible so z1 = z2 = z3 and pressure is always atmospheric p1= p2 = p3 =0. So
u1= u2 = u3 =15 m/s
By continuity Q1= Q2 + Q3
u1a1 = u2a2 + u3a3
so a1 = a2 + a3
Put the axes normal to the plate, as we know that the resultant force is normal to the plate.
Q1 = a1u = 210-315 = 0.03
Q1 = (a2 + a3) u
Q2 = a2u
Q3 = (a1 - a2)u
Calculate total force.
Component in direction of jet = 390 sin
60 = 338 N
As there is no force parallel to the plate Fty = 0
Thus 3/4 of the jet goes up, 1/4 down
6.4
A 75mm diameter jet of water having a velocity of 25m/s strikes a flat
plate, the normal of which is inclined at 30 to the jet. Find the force
normal to the surface of the plate.
[2.39kN]
From the question, djet = 0.075m u1=25m/s Q = 25p(0.075/2)2 = 0.11 m3/s
Force normal to plate is
FTx = rQ( 0 - u1x )
FTx = 10000.11 ( 0 - 25 cos 30 ) = 2.39 kN
6.5
The outlet pipe from a pump is a bend of 45 rising in the vertical plane
(i.e. and internal angle of 135). The bend is 150mm diameter at its inlet
and 300mm diameter at its outlet. The pipe axis at the inlet is horizontal
and at the outlet it is 1m higher. By neglecting friction, calculate the
force and its direction if the inlet pressure is 100kN/m2 and
the flow of water through the pipe is 0.3m3/s. The volume of
the pipe is 0.075m3.
[13.94kN at 67 40' to the horizontal]
1&2 Draw the control volume and the
axis system
p1 = 100 kN/m2, Q = 0.3 m3/s q = 45
d1 = 0.15 m d2 = 0.3 m
A1 = 0.177 m2 A2
= 0.0707 m2
3 Calculate the total force
in the x direction
by continuity 
,
so
and in the y-direction
4 Calculate the pressure force.
We know pressure at the inlet but not at the outlet.
we can use Bernoulli to calculate this
unknown pressure.
where hf is the friction loss
In the question it says this can be ignored, hf=0
The height of the pipe at the outlet is 1m above the inlet.
Taking the inlet level as the datum:
z1 = 0 z2 = 1m
So the Bernoulli equation becomes:
5 Calculate the body force
The only body force is the force due to gravity. That is the weight acting in the y direction.
There are no body forces in the x direction,
6 Calculate the resultant force
And the resultant force on the fluid is given by
And the direction of application is
The force on the bend is the same magnitude but in the opposite direction
6.6
The force exerted by a 25mm diameter jet against a flat plate normal to
the axis of the jet is 650N. What is the flow in m3/s?
[0.018 m3/s]
From the question, djet = 0.025m FTx = 650 N
Force normal to plate is
FTx = rQ( 0 - u1x )
650 = 1000Q ( 0 - u )
Q = au = (pd2/4)u
650 = -1000au2 = -1000Q2/a
650 = -1000Q2/(p0.0252/4)
Q = 0.018m3/s
6.7
A curved plate deflects a 75mm diameter jet through an angle of 45. For a
velocity in the jet of 40m/s to the right, compute the components of the
force developed against the curved plate. (Assume no friction).
[Rx=2070N, Ry=5000N down]
From the question:
Calculate the total force using the momentum equation:
Body force and pressure force are 0.
So force on vane:
6.8
A 45 reducing bend, 0.6m diameter upstream, 0.3m diameter downstream, has
water flowing through it at the rate of 0.45m3/s under a
pressure of 1.45 bar. Neglecting any loss is head for friction, calculate
the force exerted by the water on the bend, and its direction of
application.
[R=34400N to the right and down, q = 14]
1&2 Draw the control volume and the
axis system
p1 = 1.45105 N/m2, Q = 0.45 m3/s q = 45
d1 = 0.6 m d2 = 0.3 m
A1 = 0.283 m2 A2
= 0.0707 m2
3 Calculate the total force
in the x direction
by continuity 
,
so
and in the y-direction
4 Calculate the pressure force.
We know pressure at the inlet but not at the outlet.
we can use Bernoulli to calculate this unknown pressure.
where hf is the friction loss
In the question it says this can be ignored, hf=0
Assume the pipe to be horizontal
z1 = z2
So the Bernoulli equation becomes:
5 Calculate the body force
The only body force is the force due to gravity.
There are no body forces in the x or y directions,
6 Calculate the resultant force
And the resultant force on the fluid is given by
And the direction of application is
The force on the bend is the same magnitude but in the opposite direction
Information provided by: http://www.efm.leeds.ac.uk