Chapter 3 x Integral Relations for a Control Volume
211
Solution: Let the CV enclose the bucket
and jet and let it move to the right at bucket
velocity V :R, so that the jet enters the
CV at relative speed (Vj :R). Then,
¦ Fx
out mu
in
Fbucket mu
(Vj :R)] m[V
j :R]
m[
or: Fbucket
j :R) 2 U A j (Vj :R)2 ,
2m(V
and the power is P
:RFbucket
2 U A j:R(Vj :R)2
Ans.
Maximum power is found by differentiating this expression:
dP
d:
0 if :R
Vj
3
§
Ans. ¨ whence Pmax
©
8
·
U A jV3j ¸
27
¹
If there were many buckets, then the full jet mass flow would be available for work:
available
m
U A jVj , P 2 U A jVj:R(Vj :R), Pmax
1
U A j V 3j
2
at :R
Vj
2
Ans.
3.52 The vertical gate in a water channel is partially open, as in Fig. P3.52.
Assuming no change in water level and a hydrostatic pressure distribution, derive an
expression for the streamwise force Fx on one-half of the gate as a function of (U, h,
w, T, V1). Apply your result to the case of water at 20°C, V1 0.8 m/s, h 2 m, w 1.5 m,
and T 50°.
Solution: Let the CV enclose sections (1) and (2), the centerline, and the inside of the
gate, as shown. The volume flows are
V1Wh
V2 Bh, or: V2
V1
W
B
V1
1
1 sin T