P3.18
An incompressible fluid flows steadily through the rectangular duct in the figure. The exit velocity profile is given by u  u max (1 – y
2
/ b
2
)(1 – z
2
/ h
2
). (a) Does this profile satisfy the correct boundary conditions for viscous fluid flow? (b) Find an analytical expression for the volume flow Q at the exit. (c) If the inlet flow is 300 ft
3
/min, estimate u max in m/s.
Solution: (a) The fluid should not slip at any of the duct surfaces, which are defined by y   b and z   h. From our formula, we see u  0 at all duct surfaces , OK. Ans.
(a)
(b) The exit volume flow Q is defined by the integral of u over the exit plane area:
Q
  u dA
 h b
  h b u max



1
 y b
2
2



1
 z h
2
2


 dy dz
 u max
 b

 b



1
 y
2 b
2


 dy
 h

 h



1
 z
2 h
2



 max

4 b

4 h


3

3


16bhu max
9
Ans . ( )
(c) Given Q  300 ft 3 /min  0.1416 m 3 /s and b = h = 10 cm, the maximum exit velocity is
Q  0.1416 m 3

16 s 9
(0.1 m)(0.1 m) u max
, solve for u max
 7.96 m/s Ans . (c)
P3.33
In some wind tunnels the test section is perforated to suck out fluid and provide a thin viscous boundary layer. The test section wall in Fig. P3.33 contains 1200 holes of 5-mm diameter each per square meter of wall area. The suction velocity through each hole is V r  8 m/s, and the test-section entrance velocity is V 1  35 m/s. Assuming incompressible steady flow of air at 20°C, compute (a) V o , (b) V 2 , and (c) V f , in m/s.
Fig. P3.33

Area of test section =
2
 rL
  
10.053
m
2
The number of holes on the test section is
1 m
2
:1200

10.0531: N , N

12064
Q suction

NQ hole

NAV r
  
(a) Find V : Q o
 Q
1 or V o

4
(2.5) 2 

4 solve for V o
 3.58
m s
Ans . (a)
(b) Q
2
 Q
1
 Q suction
 solve for V f


4
2  
2

4
(0.8) , or: V
2
 31.2
m
Ans . (b)
(c) Find V : Q f
 Q
2 or V s f

4
(2.2) 2 

4
4.13
m s
Ans . (c)
P3.49
The horizontal nozzle in Fig. P3.49 has D 1  12 in, D 2  6 in, with p 1  38 psia and V 2
 56 ft/s. For water at 20°C, find the force provided by the flange bolts to hold the nozzle fixed.
Solution: For an open jet, p 2  p a  15 psia. Subtract p a everywhere so the only nonzero pressure is p 1  38  15  23 psig.
The mass balance yields the inlet velocity:
V A =V A
2
, V
1

4
(12) 2 

4
1
 14 ft s
The density of water is 1.94 slugs per cubic foot. Then the horizontal force balance is
 F x
  F bolts



4
2  m u  m u  m(V
2
 V )

Compute F bolts
 

4 ft s ft s
 1700 lbf Ans .
P3.61
A 20°C water jet strikes a vane on a tank with frictionless wheels, as shown. The jet turns and falls into the tank without spilling. If   30°, estimate the horizontal force F needed to hold the tank stationary.
Solution: The CV surrounds the tank and wheels and cuts through the jet, as shown. We should assume that the splashing into the tank does not increase the x-momentum of the water in the tank . Then we can write the CV horizontal force relation:
 F x
F d dt


 tank
 m u   jet

Thus F   A V 2 j


1.94 slug ft 3  
2
ft
 
50 ft s

2
 106 lbf Ans .
Fig. P3.61

P3.95
A cylindrical water tank discharges through a well-rounded orifice to hit a plate, as in Fig. P3.95.
Use the Torricelli formula of Prob. P3.81 to estimate the exit velocity. ( a ) If, at this
CV h instant, the force F required to hold the plate is 40 N, what is the depth h ? F
( b ) If the tank surface is dropping at the rate of 5 cm every 2 seconds, what is the tank diameter D ?
d = 4 cm
D
Fig. P3.95
Solution : For water take  = 998 kg/m 3 . The control volume surrounds the plate and yields

F x

F
   in u in
But Torricelli
Given data : says h

 
V
2 jet jet
(

V jet
)

2 gh ; Thus
 
A jet
V h
 jet
( V jet
)
 

4 d
2
V
2 jet
F

(

/ 4 ) d
2
( 2 g )
40 N
( 998 kg / m
3
)(

/ 4 )( 0 .
04 m )
2
( 2 )( 9 .
81 m / s
2
)

1.63
m Ans .( a )
(b) In 2 seconds, h drops from 1.63m to 1.58m, not much change. So, instead of a laborious calculus solution, find Q jet,av
for an average depth h av
= (1.63+1.58)/2 = 1.605 m:
Q av

A jet
2 gh av


4
( 0 .
04 m )
2
2 ( 9 .
81 m / s
2
)( 1 .
605 m )

0 .
00705 m
3
/ s
Equate Q
 t

A tank
 h , or : D

(

Q
 t
/ 4 )
 h

( 0 .
00705 )( 2 s )
(

/ 4 )( 0 .
05 m )

0.60
m Ans .( b )
P3.131 In Fig. P3.131 both fluids are at 20°C. If V 1  1.7 ft/s and losses are neglected, what should the manometer reading h ft be?
Solution: By continuity, establish V 2 :

A V

A V
2
, V
2
 V (D /D ) 2  1.7(3/1) 2  15.3 ft s
Now apply Bernoulli between 1 and 2 to establish the pressure at section 2: p
1


2
V 2
1
  gz
1
 p
2


2
V 2
2
 
Fig. P3.131 or: p
1
 (1.94/2)(1.7) 2    2  (62.4)(10), p
1
 848 psf
This is gage pressure. Now the manometer reads gage pressure, so p
1
 p a
 848 lbf ft 2
 (  merc
  water
 1.08 ft Ans .
P3.137
In Fig. P3.137 the piston drives water at 20°C. Neglecting losses, estimate the exit velocity V 2 ft/s. If D 2 is further constricted, what is the maximum possible value of V 2 ?
Fig. P3.137
Solution: Find p 1 from a freebody of the piston:

F x
 
1
 p A , or: p
1
 p a

F


10.0 lbf
A
1
( /4)(8/12)
2

28.65 lbf ft
2
Now apply continuity and Bernoulli from 1 to 2:

V A  V A , or V
1

1
4 p
1


V
2
2
1  p a


V
2
2
2
V
2
2

2(28.65)
, V
2

5.61
ft s
Ans .
If we reduce section 2 to a pinhole, V 2 will drop off slowly until V 1 vanishes:
Severely constricted section 2: V
2

2(28.65)
 5.43
ft s
Ans .
P3.155
The centrifugal pump of Fig. P3.155 has a flow rate Q and exits the impeller at an angle  2 relative to the blades, as shown. The fluid enters axially at section 1. Assuming incompressible flow at shaft angular velocity  , derive a formula for the power P required to drive the impeller.
Solution: Relative to the blade, the fluid exits at velocity V rel,2 tangent to the blade, as shown in Fig. P3.116. But the Euler turbine formula, Ans. (a) from Example 3.18 of the text,
Torque T 



Q(r V  r V )
Qr V (assuming V t1
 0) involves the absolute fluid velocity tangential to the blade circle (see Fig. 3.15). To derive this velocity we need the “velocity diagram” shown above, where absolute exit velocity V 2 is found by adding blade tip rotation speed  r 2 to V rel,2 . With trigonometry,
V t2
 r
2
   n2
 Q/A exit


Q is the normal velocity
With torque T known, the power required is P  T  The final formula is:
P   Qr
2


 r
2
 



Q 
 cot 
2


Ans .
Fig. P3.155

P3.169
When the pump in Fig. P3.169 draws 220 m
3
/h of water at 20  C from the reservoir, the total friction head loss is 5 m. The flow discharges through a nozzle to the atmosphere Estimate the pump power in kW delivered to the water.
Solution: Let “1” be at the reservoir surface and “2” be at the nozzle exit, as shown. We need to know the exit velocity:
V
2
 Q/A
2


220/3600
(0.025) 2
 31.12 m s
, while V
1
 0 (reservoir surface)
Now apply the steady flow energy equation from (1) to (2): p

1 
V 2
1 g 2g
  p

2 
V g 2g
2
2  z
2
 h f
 h ,
       p
 56.4 m.
The pump power P   gQh p  (998)(9.81)(220/3600)(56.4)
 33700 W  33.7 kW Ans .
Fig. P3.169
P3.180
Water at 20  C is pumped at 1500 gal/ min from the lower to the upper reservoir, as in Fig.
P3.180. Pipe friction losses are approximated by h f  27 V 2 /(2 g ), where V is the average velocity in the pipe. If the pump is 75 percent efficient, what horse-power is needed to drive it?
Solution: First evaluate the average velocity in the pipe and the friction head loss:
Q 
1500
448.8
 3.34 ft 3
, so V s

Q

A
3.34
 (3/12) 2
 17.0 ft s
(17.0) 2 and h f
 27
2(32.2)
 121 ft
Then apply the steady flow energy equation: p

1 g

V 2
1
2g
 z
1
 p

2 g

V
2g
2
2  z
2
 h f
 h , p

Thus h p
 221 ft, so P pump

 Qh
 p 
(62.4)(3.34)(221)
0.75
 61600 s
 112 hp Ans .
Fig. P3.180
C3.1
In a certain industrial process, oil of density  flows through the inclined pipe in the figure. A
U-tube manometer with fluid density  m , measures the pressure difference between points 1 and 2, as shown. The flow is steady, so that fluids in the U-tube are stationary. (a) Find an analytic expression for p 1  p 2 in terms of system parameters. (b) Discuss the conditions on h necessary for there to be no flow in the pipe. (c) What about flow up , from 1 to 2? (d) What about flow down , from
2 to 1?
Solution: (a) Start at 1 and work your way around the U-tube to point 2: or : p
1
  gs   gh   m p
1
 p
2
  g z (  m
  gh   gs     p
2
) gh
, where  
2
 z
1
Ans.
(a)
(b) If there is no flow, the pressure is entirely hydrostatic, therefore  p   g and, since  m   , it follows from Ans. (a) above that h  0 Ans.
(b)
(c) If h is positive (as in the figure above), p 1 is greater than it would be for no flow, because of head losses in the pipe. Thus, if h  0, flow is up from 1 to 2 . Ans. (c)
(d) If h is negative, p 1 is less than it would be for no flow, because the head losses act against hydrostatics. Thus, if h  0, flow is down from 2 to 1 . Ans. (d)
Note that h is a direct measure of flow, regardless of the angle  of the pipe.