Problem set #6
1) Water flows by gravity from one lake to another as sketched in Figure 1 at the steady
rate of 82 gpm.
(a) What is the loss in available energy associated with this flow?
(b) If this same amount of loss is associated with pumping the fluid from the lower lake
to the higher one at the same flowrate, estimate the amount of pumping power required.
Figure 1
energy loss = gΔz = 32.17
ft
ft lb
 47 ft = 1511
2
s
slug
(b) Energy required to pump at the same rate counter to this flow.
Ws = 1.94
slug
ft 3 
ft lb 
ft lb

0.182
32.17  47+1511
=1067


3
ft
sec 
slug 
sec
2) Water is to be pumped from the large tank shown in Figure with an exit velocity of 5.5
m/s. It was determined that the original pump (pump 1) that supplies 1 kW of power to
the water did not produce the desired velocity. Hence, it is proposed that an additional
pump (pump 2) be installed as indicated to increase the flowrate to the desired value.
How much power must pump 2 add to the water? The head loss for this flow is hL =
250Q2, where hL is in m when Q is in m3/s.
Wp =Wp1 +Wp2  Wp2 =Wp  Wp1 = 2.258 kW 1 kW = 1.258 kW
3) Oil (SAE 30) at 15.6oC flows steadily between fixed, horizontal, parallel plates. The
pressure drop per unit length along the channel is 35 kPa/m, and the distance between the
plates is 4 mm. The flow is laminar. Determine: (a) the volume rate of flow (per meter of
width), (b) the magnitude of the shearing stress acting on the bottom plate, and (c) the
velocity along the centerline of the channel.
2 .002 m 
N
m2
q=
 35000 3 = 4.912 104
N s 
m
s

3  0.38 2 
m 

3
(b)
Tyx =
p
Δp
N
N
y=
h = 35000 3  .002 m = 70 2
x
l
m
m
(c)
3
q
3 q 3
vmax = V,V =
 vmax =
=
2
2h
2 2h 2
m2
s = 0.372 m
2  .002 m
s
4.912 104
4) Two fixed, horizontal, parallel plates are spaced 0.6 in. apart. A viscous liquid (μ = 8 ×
10-3 lb•s/ft2, SG = 0.9) flows between the plates with a mean velocity of 0.6 ft/s. The flow
is laminar. (a) Determine the pressure drop per foot in the direction of flow. (b) What is
the maximum velocity in the channel?
(a)
Δp 3μv
= 2 =
l
h
lb  s
ft
 .6
2
ft
s = 23.04 lb
2
ft 3
 .3 
f
t


 12 
3  8 103
(b)
3
3 ft
ft
vmax = V = .6 = .9
2
2 s
s
5) A viscous fluid (specific weight = 80 lb/ft3; viscosity = 0.03 lb•s/ft2) is contained
between two infinite, horizontal parallel plates as shown in the Figure. The fluid moves
between the plates under the action of a pressure gradient, and the upper plate moves with
a velocity U while the bottom plate is fixed. A U-tube manometer connected between two
points along the bottom indicates a differential reading of 0.1 in. If the upper plate moves
with a velocity of 0.03 ft/s, at what distance from the bottom plate does the maximum
velocity in the gap between the two plates occur? Assume laminar flow.
lb  s
ft
1
.03
ft
2
ft
s + 12 = .0741 ft  .0741 12 ft = .8887 in
yumax =
1
lb 
2
1 in

ft   .3334 3 
12
ft 

6) A vertical shaft passes through a bearing and is lubricated with an oil having a
viscosity of 0.2 N•s/m2 as shown in the Figure. Assume that the flow characteristics in
the gap between the shaft and bearing are the same as those for laminar flow between
infinite parallel plates with zero pressure gradient in the direction of flow. Estimate the
torque required to overcome viscous resistance when the shaft is turning at 90 rev/min.
.03
l

 .075 
T = 2πri3  μω  = 2π 
m
b

 2

3
.160 m
 N s 
= 0.399 Nm
 .2 2  3π
-3
 m  .25×10 m
7) A viscous liquid (μ = 0.012 lb•s/ft2, ρ = 1.79 slugs/ft3) flows through the annular space
between two horizontal, fixed, concentric cylinders. If the radius of the inner cylinder is
3.5 in. and the radius of the outer cylinder is 4.5 in., what is the pressure drop along the
axis of the annulus per foot when the volume flowrate is 0.15 ft3/s?
8μQ
Δp
π
=
2
l
ro2  ri 2 

4
4
ro  ri 
r
ln o
ri
lb  s
ft 3

.15
ft 2
sec
lb
π
=
= 17.80 2
2
2 2
ft
 4.5   3.5  
ft  - 
ft  

4
4
 12   12  
 4.5 
 3.5 

 ft- 
 ft4.5 in
 12 
 12 
ln
3.5 in
8  .012
8) A wire of diameter d is stretched along the centerline of a pipe of diameter D. For a
given pressure drop per unit length of pipe, by how much does the presence of the wire
reduce the flowrate if (a) d/D = 0.4; (b) d/D = 0.05?
(a)
2

1  .4 2  
4

  = 0.7957 = 79.57%
flow reduction = 1  1  .4  + 

ln .4  



(b)
2

1  .052  
4

  = 0.3321 = 33.21%
flow reduction = 1  1  .05  + 

ln .05 


