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Solutions Manual
for
Fundamentals of Thermal Fluid Sciences
5th Edition
Yunus A. Çengel, John M. Cimbala, Robert H. Turner
McGraw-Hill, 2017
Chapter 14
INTERNAL FLOW
PROPRIETARY AND CONFIDENTIAL
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without the prior written permission of McGraw-Hill Education.
14-2
Laminar and Turbulent Flow
14-1C
Solution We are to compare pipe flow in air and water.
speed, the Reynolds number will be higher for water flow, and thus the flow is more likely to be turbulent for water.
Discussion The actual viscosity (dynamic viscosity)
is larger for water than for air, but the density of water is so
much greater than that of air that the kinematic viscosity of water ends up being smaller than that of air.
14-2C
Solution We are to compare the wall shear stress at the inlet and outlet of a pipe.
zero, and decreases gradually to the fully developed value. The same is true for turbulent flow.
Discussion We are assuming that the entrance is well-rounded so that the inlet flow is nearly uniform.
14-3C
Solution We are to define and discuss hydraulic diameter.
14-4C
Solution We are to define and discuss hydrodynamic entry length.
14-5C
Solution We are to discuss why pipes are usually circular in cross section.
14-6C
Solution We are to define and discuss Reynolds number for pipe and duct flow.
14-4
14-7C
14-8C
Solution We are to express the Reynolds number for a circular pipe in terms of mass flow rate.
Analysis Reynolds number for flow in a circular tube of diameter D is expressed as
4 and
m m m
V
14-11E
Solution We are to estimate the Reynolds number for flow through a
pipe, and determine if it is laminar or turbulent.
Assumptions 1 The water is at 20oC. 2 The discharge area is perfectly
round (we ignore the rim effects – there appear to be some protrusions
around the rim – three of them are visible in the picture).
Analysis We use the people to estimate the diameter of the pipe.
Assuming the guy in the blue shirt (who by the way is Secretary of the
Interior Dirk Kempthorne) is six feet tall, the pipe diameter is about 13.8 ft.
14-6
Fully Developed Flow in Pipes
14-12C
works for fully developed laminar pipe flow in round pipes since
cc AVAV )2/( maxavg
V
.
Discussion This is not true for turbulent flow, so one must be careful that the flow is laminar before trusting this
measurement. It is also not true if the pipe is not round, even if the flow is fully developed and laminar.
14-13C
Solution We are to examine a claim about volume flow rate in laminar pipe flow.
4
3
1)2/( max
2/
2
2
max
V
R
r
VRV
Rr
, which is much larger than Vmax/2.
Discussion There is, of course, a radial location in the pipe at which the local velocity is equal to the average velocity.
Can you find that location?
Solution We are to discuss the value of shear stress at the center of a pipe.
14-15C
Solution We are to discuss whether the maximum shear stress in a turbulent pipe flow occurs at the wall.
14-16C
Solution We are to discuss how the wall shear stress varies along the flow direction in a pipe.
14-17C
Solution We are to discuss the velocity profile in fully developed pipe flow.
14-18C
Solution We are to discuss the relationship between friction factor and pressure loss in pipe flow.
Analysis The friction factor for flow in a tube is proportional to the pressure loss. Since the pressure loss along
14-8
14-19C
Solution We are to discuss whether fully developed pipe flow is one-, two-, or three-dimensional.
Analysis The geometry is axisymmetric, which is two-dimensional. However, since the velocity profile does not
14-20C
Solution We are to discuss the change in head loss when the pipe length is doubled.
14-21C
Solution We are to compare the head loss when the pipe diameter is halved.
Analysis In fully developed laminar flow in a circular pipe, the head loss is given by
g
V
D
L
Dg
V
D
L
DVg
V
D
L
g
V
D
L
fhL2
64
2/
64
2Re
64
2
222
The average velocity can be expressed in terms of the flow rate as
4/
2
D
A
V
c
VV
. Substituting,
42222
128
2
464
4/
2
64
Dg
L
Dg
L
DD
g
L
D
hL
VVV
Therefore, at constant flow rate and pipe length, the head loss is inversely proportional to the 4th power of diameter, and
thus reducing the pipe diameter by half increases the head loss by a factor of 16.
Discussion This is a very significant increase in head loss, and shows why larger diameter tubes lead to much smaller
pumping power requirements.
14-22C
Solution We are to explain why friction factor is independent of Re at very large Re.
14-23C
Solution We are to define and discuss turbulent viscosity.
14-24C
Solution We are to discuss the change in head loss due to a decrease in viscosity by a factor of two.
Analysis In fully developed laminar flow in a circular pipe, the pressure loss and the head loss are given by
2
32
D
LV
PL
and
2
32
gD
LV
g
P
hL
L
When the flow rate and thus the average velocity are held constant, the head loss becomes proportional to viscosity.
Therefore, the head loss is reduced by half when the viscosity of the fluid is reduced by half.
Discussion This result is not valid for turbulent flow – only for laminar flow. It is also not valid for laminar flow in
situations where the entrance length effects are not negligible.
14-10
14-25C
Solution We are to discuss the relationship between head loss and pressure drop in pipe flow.
14-26C
Solution We are to discuss if the friction factor is zero for laminar pipe flow with a perfectly smooth surface.
14-27C
Solution We are to discuss why the friction factor is higher in turbulent pipe flow compared to laminar pipe flow.
h
y
U
14-28
Solution The velocity profile for the flow of a fluid between two large parallel plates is given. A relation for the flow
rate through the plates is to be determined.
Assumptions 1 The flow is steady and incompressible.
Analysis
1
0
2
0
2
0
131
2
3
2)(2)()( h
y
d
h
y
Ubhdy
h
y
Ub
bdyyUbdyyUdAyU
h
h
hh
V
hy
hy
h
y
h
y
Ubh
/1
/0
3
3
1
3
V
UbhUbhUbh 23
3
2
0
3
1
1
V
14-12
14-29
Solution Water flows in a reducing pipe section. The flow upstream is laminar and the flow downstream is turbulent.
The ratio of centerline velocities is to be determined.
14-30
Solution The average flow velocity in a pipe is given. The pressure drop, the head loss, and the pumping power are
to be determined.
14-31
Solution Air enters the constant spacing between the glass cover and the plate of a solar collector. The pressure drop
of air in the collector is to be determined.
14-15
14-32E
Solution The flow rate and the head loss in an air duct is given. The minimum diameter of the duct is to be
0.04615 lbm/fth, and
= 0.6512 ft2/s = 1.80910-4 ft2/s.
Analysis The average velocity, Reynolds number, friction factor, and the head loss relations can be expressed as (D is
in ft, V is in ft/s, Re and f are dimensionless)
s
VDVD
D
s
D
A
V
c
/ft 10809.1
Re
4/
/ft 12
4/
24
2
3
2
VV
ff
D
fRe
51.2
log0.2
Re
51.2
7.3
/
log0.2
1
ft 400
222 V
V
L
V
L
L = 400 ft
D
Air
12 ft3/s
14-33
Solution In fully developed laminar flow in a circular pipe, the velocity at r = R/2 is measured. The velocity at the
2
2
max 1)( R
r
uru
where umax is the maximum velocity which occurs at pipe center, r = 0. At r =R/2,
4
3
4
1
1
)2/(
1)2/( max
max
2
2
max
u
u
R
R
uRu
Solving for umax and substituting,
m/s 14.7 3
m/s) 11(4
3
)2/(4
max
Ru
u
14-34
Solution The velocity profile in fully developed laminar flow in a circular pipe is given. The average and maximum
2
2
max 1)( R
r
uru
The velocity profile in this case is given by
)/1(4)( 22 Rrru
r
0
u(r) = umax(1-r2/R2)
R
r
u(r) = umax(1-r2/R2)
R
14-17
14-35
Solution The velocity profile in fully developed laminar flow in a circular pipe is given. The average and maximum
2
2
max 1)( R
r
uru
= 4.00 m/s. Then the average velocity and volume flow rate become
4 m/s
22
max
avg
u
V 2.00 m/s
/sm 0.0308 3
]m) (0.07m/s)[ (2)( 22
RVAV avgcavg
V
Discussion Compared to the previous problem, the average velocity remains the same since the maximum velocity (at
the centerline) remains the same, but the volume flow rate increases as the diameter increases.
0
u(r) = umax(1-r2/R2)
14-36
Solution The flow rate through a specified water pipe is given. The pressure drop, the head loss, and the pumping
power requirements are to be determined.
14-19
14-37
Solution Laminar flow through a square channel is considered. The change in the head loss is to be determined when
14-20
14-38
Solution Turbulent flow through a smooth pipe is considered. The change in the head loss is to be determined when
2.0
Re184.0
f
where
VD
Re
Then the head loss of the fluid for turbulent flow can be expressed as
g
V
D
L
D
g
V
D
L
VD
g
V
D
L
g
V
D
L
fhL2
184.0
2
184.0
2
Re184.0
2
8.1
2.0
2
2.0
2
2.0
2
1,
which shows that the head loss is proportional to the 1.8th power of the average velocity. Therefore, the head loss increases
by a factor of 21.8 = 3.48 when the average velocity is doubled. This can also be shown as
1,1,
8.1
8.1
2.0
8.1
8.1
2.0
8.1
2
2.0
2,
48.32
2
184.02
2
)2(
184.0
2
184.0
LL
L
hh
g
V
D
LD
g
V
D
LD
g
V
D
LD
h
For fully rough flow in a rough pipe, the friction factor is independent of the Reynolds
V
