Chapter 3 x Integral Relations for a Control Volume
185
3.15 Water flows steadily through the
round pipe in the figure. The entrance
velocity is Vo. The exit velocity
approximates turbulent flow, u
umax(1 r/R)1/7. Determine the ratio
Uo/umax for this incompressible flow.
Solution: Inlet and outlet flow must balance:
R
Q1
Q2 , or:
³ Uo 2S r dr
0
R
³
0
1/7
r·
§
umax ¨ 1 ¸ 2S r dr , or: U oS R 2
© R¹
umax
Cancel and rearrange for this assumed incompressible pipe flow:
49
60
Uo
umax
Ans.
3.16 An incompressible fluid flows past
an impermeable flat plate, as in Fig. P3.16,
with a uniform inlet profile u Uo and a
cubic polynomial exit profile
§ 3K K 3 ·
y
u | Uo ¨
¸ where K
Fig. P3.16
G
© 2 ¹
Compute the volume flow Q across the top surface of the control volume.
Solution: For the given control volume and incompressible flow, we obtain
G
0
Q top Q right Q left
G
§ 3 y y3 ·
3 ¸ b dy ³ U o b dy
Q ³ Uo ¨
G
2
2G ¹
©
0
0
5
Q U o bG U o bG , solve for Q
8
3.17 Incompressible steady flow in the
inlet between parallel plates in Fig. P3.17 is
uniform, u Uo 8 cm/s, while downstream
the flow develops into the parabolic laminar
profile u az(zo z), where a is a constant.
If zo 4 cm and the fluid is SAE 30 oil at
20qC, what is the value of umax in cm/s?
3
U o bG
8
Ans.
Fig. P3.17
49S 2
R
60