Mechanics of Fluids

156 The momentum equation
✷
(b) F = ϱAu2 (u1 − u4) = ϱA (u1 + u4)
2
(u1 − u4)
= 1.2 kg · m −3 × π
4 × (12 m)2 (20 + 8) m · s−1
× × 12 m · s
2
−1
= 22.8 × 10 3 N
�
u2 1
(c) P = ϱQ
2 − u2 � �
4
u2 1
= ϱAu2
2
= ϱA (u1
�
+ u4) u2 1
2 2 − u2 �
4
2
2 − u2 4
2
= 1.2 kg · m −3 × π
4 (12 m)2 (20 + 8) m · s−1
×
2
��
20 m · s−1 �2 �
8m· s−1 �2 �
×
2
−
= 319 000 W = 319 kW
PROBLEMS
2
4.1 A stationary curved vane deflects a 50 mm diameter jet of water
through 150 ◦ . Because of friction over the surface, the water
leaving the vane has only 80% of its original velocity. Calculate
the volume flow rate necessary to produce a fluid-dynamic force
of 2000 N on the vane.
4.2 The diameter of a pipe-bend is 300 mm at inlet and 150 mm at
outlet and the flow is turned through 120 ◦ in a vertical plane.
The axis at inlet is horizontal and the centre of the outlet section
is 1.4 m below the centre of the inlet section. The total volume
of fluid contained in the bend is 0.085 m 3 . Neglecting friction,
calculate the magnitude and direction of the net force exerted
on the bend by water flowing through it at 0.23 m 3 · s −1 when
the inlet gauge pressure is 140 kPa.
4.3 Air at constant density 1.22 kg · m −3 flows in a duct of internal
diameter 600 mm and is discharged to atmosphere. At the outlet
end of the duct, and coaxial with it, is a cone with base diameter
750 mm and vertex angle 90 ◦ . Flow in the duct is controlled
by moving the vertex of the cone into the duct, the air then
escaping along the sloping sides of the cone. The mean velocity
in the duct upstream of the cone is 15 m · s −1 and the air leaves
the cone (at the 750 mm diameter) with a mean velocity of
60 m · s −1 parallel to the sides. Assuming temperature changes
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