fluid_mechanics

214 Chapter 5 ■ Finite Control Volume Analysis
SOLUTION
To determine the magnitude and direction of the force, F, exerted
by the water on the vane, we apply Eq. 5.29 to the contents of the
moving control volume shown in Fig. E5.17b. The forces acting
on the contents of this control volume are indicated in
Fig. E5.17c. Note that since the ambient pressure is atmospheric,
all pressure forces cancel each other out. Equation 5.29 is applied
to the contents of the moving control volume in component
directions. For the x direction 1positive to the right2, we get
or
where
or
cs
W x r W nˆ dA R x
1W 1 21m # 12 1W 2 cos 45°21m # 22 R x
m # 1 r 1 W 1 A 1 and m # 2 r 2 W 2 A 2 .
For the vertical or z direction 1positive up2 we get
cs
W z rW nˆ dA R z w w
1W 2 sin 45°21m # 22 R z w w
We assume for simplicity that the water flow is frictionless and that
the change in water elevation across the vane is negligible. Thus,
from the Bernoulli equation 1Eq. 3.72 we conclude that the speed of
the water relative to the moving control volume, W, is constant or
W 1 W 2
The relative speed of the stream of water entering the control volume,
W 1 , is
W 1 V 1 V 0 100 fts 20 fts 80 fts W 2
The water density is constant so that
r 1 r 2 1.94 slugsft 3
Application of the conservation of mass principle to the contents
of the moving control volume 1Eq. 5.162 leads to
m # 1 r 1 W 1 A 1 r 2 W 2 A 2 m # 2
(1)
(2)
Combining results we get
or
Also,
where
Thus,
R x 11.94 slugsft 3 2180 fts2 2 10.006 ft 2 211 cos 45°2
21.8 lb
R x rW 2 1 A 1 11 cos 45°2
R z rW 1 2 1sin 45°2A 1 w w
w w rgA 1 /
R z 11.94 slugsft 3 2180 fts2 2 1sin 45°210.006 ft 2 2
162.4 lbft 3 210.006 ft 2 211 ft2
52.6 lb 0.37 lb 53 lb
Combining the components we get
R 2R x 2 R z 2 3121.8 lb2 2 153 lb2 2 4 1 2 57.3 lb
The angle of R from the x direction, a, is
a tan 1 R z
R x
tan 1 153 lb21.8 lb2 67.6°
The force of the water on the vane is equal in magnitude but opposite
in direction from R; thus it points to the right and down at
an angle of 67.6° from the x direction and is equal in magnitude
to 57.3 lb.
(Ans)
COMMENT The force of the fluid on the vane in the x-
direction, R x 21.8 lb, is associated with x-direction motion of the
vane at a constant speed of 20 fts. Since the vane is not accelerating,
this x-direction force is opposed mainly by a wheel friction
force of the same magnitude. From basic physics we recall that the
power this situation involves is the product of force and speed. Thus,
p R x V 0
121.8 lb2120 ft s2
5501ft lb21hp s2
0.79 hp
All of this power is consumed by friction.
It is clear from the preceding examples that a flowing fluid can be forced to
1. change direction
2. speed up or slow down
3. have a velocity profile change
4. do only some or all of the above
5. do none of the above
A net force on the fluid is required for achieving any or all of the first four above. The forces
on a flowing fluid balance out with no net force for the fifth.
Typical forces considered in this book include
(a) pressure
(b) friction
(c) weight