Page 1
Chapter 8 Flow in Pipes
Chapter 8
FLOW IN PIPES
Laminar and Turbulent Flow
81C Liquids are usually transported in circular pipes because pipes with a circular crosssection can
withstand large pressure differences between the inside and the outside without undergoing any significant
distortion.
82C Reynolds number is the ratio of the inertial forces to viscous forces, and it serves as a criteria for
determining the flow regime. At large Reynolds numbers, for example, the flow is turbulent since the
inertia forces are large relative to the viscous forces, and thus the viscous forces cannot prevent the random
and rapid fluctuations of the fluid. It is defined as follows:
(a) For flow in a circular tube of inner diameter D:
?
VD
= Re
(b) For flow in a rectangular duct of crosssection a × b:
?
h
VD
= Re
where D
A
p
ab
ab
ab
ab
h
c
==
+
=
+
4 4
2
2
() ( )
is the hydraulic diameter.
81
83C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
noting that Re = VD/ ?, the Reynolds number is higher for motion in water for the same diameter and speed.
a
D
b
84C Reynolds number for flow in a circular tube of diameter D is expressed as
?
VD
= Re where
?
µ
?
?p p ? ?
= = = = = and
4
) 4 / (
2 2
D
m
D
m
A
m
V
c
avg
& & &
V
Substituting,
m, V
µ p ? µ ?p ? D
m
D
D m VD & & 4
) / (
4
Re
2
= = =
85C Engine oil requires a larger pump because of its much larger density.
86C The generally accepted value of the Reynolds number above which the flow in a smooth pipe is
turbulent is 4000.
87C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
for the same diameter and speed, the Reynolds number will be higher for water flow, and thus the flow is
more likely to be turbulent for water.
88C For flow through noncircular tubes, the Reynolds number and the friction factor are based on the
hydraulic diameter D
h
defined as
p
A
D
c
h
4
= where A
c
is the crosssectional area of the tube and p is its
perimeter. The hydraulic diameter is defined such that it reduces to ordinary diameter D for circular tubes
since D
D
D
p
A
D
c
h
= = =
p
p 4 / 4 4
2
.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
Page 2
Chapter 8 Flow in Pipes
Chapter 8
FLOW IN PIPES
Laminar and Turbulent Flow
81C Liquids are usually transported in circular pipes because pipes with a circular crosssection can
withstand large pressure differences between the inside and the outside without undergoing any significant
distortion.
82C Reynolds number is the ratio of the inertial forces to viscous forces, and it serves as a criteria for
determining the flow regime. At large Reynolds numbers, for example, the flow is turbulent since the
inertia forces are large relative to the viscous forces, and thus the viscous forces cannot prevent the random
and rapid fluctuations of the fluid. It is defined as follows:
(a) For flow in a circular tube of inner diameter D:
?
VD
= Re
(b) For flow in a rectangular duct of crosssection a × b:
?
h
VD
= Re
where D
A
p
ab
ab
ab
ab
h
c
==
+
=
+
4 4
2
2
() ( )
is the hydraulic diameter.
81
83C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
noting that Re = VD/ ?, the Reynolds number is higher for motion in water for the same diameter and speed.
a
D
b
84C Reynolds number for flow in a circular tube of diameter D is expressed as
?
VD
= Re where
?
µ
?
?p p ? ?
= = = = = and
4
) 4 / (
2 2
D
m
D
m
A
m
V
c
avg
& & &
V
Substituting,
m, V
µ p ? µ ?p ? D
m
D
D m VD & & 4
) / (
4
Re
2
= = =
85C Engine oil requires a larger pump because of its much larger density.
86C The generally accepted value of the Reynolds number above which the flow in a smooth pipe is
turbulent is 4000.
87C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
for the same diameter and speed, the Reynolds number will be higher for water flow, and thus the flow is
more likely to be turbulent for water.
88C For flow through noncircular tubes, the Reynolds number and the friction factor are based on the
hydraulic diameter D
h
defined as
p
A
D
c
h
4
= where A
c
is the crosssectional area of the tube and p is its
perimeter. The hydraulic diameter is defined such that it reduces to ordinary diameter D for circular tubes
since D
D
D
p
A
D
c
h
= = =
p
p 4 / 4 4
2
.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
Chapter 8 Flow in Pipes
89C The region from the tube inlet to the point at which the boundary layer merges at the centerline is
called the hydrodynamic entrance region, and the length of this region is called hydrodynamic entry length.
The entry length is much longer in laminar flow than it is in turbulent flow. But at very low Reynolds
numbers, L
h
is very small (L
h
= 1.2D at Re = 20).
810C The wall shear stress t
w
is highest at the tube inlet where the thickness of the boundary layer is zero,
and decreases gradually to the fully developed value. The same is true for turbulent flow.
811C In turbulent flow, the tubes with rough surfaces have much higher friction factors than the tubes
with smooth surfaces, and thus much larger pressure drop. In the case of laminar flow, the effect of surface
roughness on the friction factor and pressure drop is negligible.
Fully Developed Flow in Pipes
812C The wall shear stress t
w
remains constant along the flow direction in the fully developed region in
both laminar and turbulent flow.
813C The fluid viscosity is responsible for the development of the velocity boundary layer.
814C In the fully developed region of flow in a circular pipe, the velocity profile will NOT change in the
flow direction.
815C The friction factor for flow in a tube is proportional to the pressure loss. Since the pressure loss
along the flow is directly related to the power requirements of the pump to maintain flow, the friction factor
is also proportional to the power requirements to overcome friction. The applicable relations are
?
?
L
L
P m
W
V
D
L
f P
?
= = ?
&
&
pump
2
and
2
816C The shear stress at the center of a circular tube during fully developed laminar flow is zero since the
shear stress is proportional to the velocity gradient, which is zero at the tube center.
817C Yes, the shear stress at the surface of a tube during fully developed turbulent flow is maximum since
the shear stress is proportional to the velocity gradient, which is maximum at the tube surface.
818C In fully developed flow in a circular pipe with negligible entrance effects, if the length of the pipe is
doubled, the head loss will also double (the head loss is proportional to pipe length).
819C Yes, the volume flow rate in a circular pipe with laminar flow can be determined by measuring the
velocity at the centerline in the fully developed region, multiplying it by the crosssectional area, and
dividing the result by 2 since .
c c
A V A V ) 2 / (
max avg
= = V
&
820C No, the average velocity in a circular pipe in fully developed laminar flow cannot be determined by
simply measuring the velocity at R/2 (midway between the wall surface and the centerline). The average
velocity is V
max
/2, but the velocity at R/2 is
4
3
1 ) 2 / (
max
2 /
2
2
max
V
R
r
V R V
R r
=
?
?
?
?
?
?
?
?
 =
=
, which is much larger than V
max
/2.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
82
Page 3
Chapter 8 Flow in Pipes
Chapter 8
FLOW IN PIPES
Laminar and Turbulent Flow
81C Liquids are usually transported in circular pipes because pipes with a circular crosssection can
withstand large pressure differences between the inside and the outside without undergoing any significant
distortion.
82C Reynolds number is the ratio of the inertial forces to viscous forces, and it serves as a criteria for
determining the flow regime. At large Reynolds numbers, for example, the flow is turbulent since the
inertia forces are large relative to the viscous forces, and thus the viscous forces cannot prevent the random
and rapid fluctuations of the fluid. It is defined as follows:
(a) For flow in a circular tube of inner diameter D:
?
VD
= Re
(b) For flow in a rectangular duct of crosssection a × b:
?
h
VD
= Re
where D
A
p
ab
ab
ab
ab
h
c
==
+
=
+
4 4
2
2
() ( )
is the hydraulic diameter.
81
83C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
noting that Re = VD/ ?, the Reynolds number is higher for motion in water for the same diameter and speed.
a
D
b
84C Reynolds number for flow in a circular tube of diameter D is expressed as
?
VD
= Re where
?
µ
?
?p p ? ?
= = = = = and
4
) 4 / (
2 2
D
m
D
m
A
m
V
c
avg
& & &
V
Substituting,
m, V
µ p ? µ ?p ? D
m
D
D m VD & & 4
) / (
4
Re
2
= = =
85C Engine oil requires a larger pump because of its much larger density.
86C The generally accepted value of the Reynolds number above which the flow in a smooth pipe is
turbulent is 4000.
87C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
for the same diameter and speed, the Reynolds number will be higher for water flow, and thus the flow is
more likely to be turbulent for water.
88C For flow through noncircular tubes, the Reynolds number and the friction factor are based on the
hydraulic diameter D
h
defined as
p
A
D
c
h
4
= where A
c
is the crosssectional area of the tube and p is its
perimeter. The hydraulic diameter is defined such that it reduces to ordinary diameter D for circular tubes
since D
D
D
p
A
D
c
h
= = =
p
p 4 / 4 4
2
.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
Chapter 8 Flow in Pipes
89C The region from the tube inlet to the point at which the boundary layer merges at the centerline is
called the hydrodynamic entrance region, and the length of this region is called hydrodynamic entry length.
The entry length is much longer in laminar flow than it is in turbulent flow. But at very low Reynolds
numbers, L
h
is very small (L
h
= 1.2D at Re = 20).
810C The wall shear stress t
w
is highest at the tube inlet where the thickness of the boundary layer is zero,
and decreases gradually to the fully developed value. The same is true for turbulent flow.
811C In turbulent flow, the tubes with rough surfaces have much higher friction factors than the tubes
with smooth surfaces, and thus much larger pressure drop. In the case of laminar flow, the effect of surface
roughness on the friction factor and pressure drop is negligible.
Fully Developed Flow in Pipes
812C The wall shear stress t
w
remains constant along the flow direction in the fully developed region in
both laminar and turbulent flow.
813C The fluid viscosity is responsible for the development of the velocity boundary layer.
814C In the fully developed region of flow in a circular pipe, the velocity profile will NOT change in the
flow direction.
815C The friction factor for flow in a tube is proportional to the pressure loss. Since the pressure loss
along the flow is directly related to the power requirements of the pump to maintain flow, the friction factor
is also proportional to the power requirements to overcome friction. The applicable relations are
?
?
L
L
P m
W
V
D
L
f P
?
= = ?
&
&
pump
2
and
2
816C The shear stress at the center of a circular tube during fully developed laminar flow is zero since the
shear stress is proportional to the velocity gradient, which is zero at the tube center.
817C Yes, the shear stress at the surface of a tube during fully developed turbulent flow is maximum since
the shear stress is proportional to the velocity gradient, which is maximum at the tube surface.
818C In fully developed flow in a circular pipe with negligible entrance effects, if the length of the pipe is
doubled, the head loss will also double (the head loss is proportional to pipe length).
819C Yes, the volume flow rate in a circular pipe with laminar flow can be determined by measuring the
velocity at the centerline in the fully developed region, multiplying it by the crosssectional area, and
dividing the result by 2 since .
c c
A V A V ) 2 / (
max avg
= = V
&
820C No, the average velocity in a circular pipe in fully developed laminar flow cannot be determined by
simply measuring the velocity at R/2 (midway between the wall surface and the centerline). The average
velocity is V
max
/2, but the velocity at R/2 is
4
3
1 ) 2 / (
max
2 /
2
2
max
V
R
r
V R V
R r
=
?
?
?
?
?
?
?
?
 =
=
, which is much larger than V
max
/2.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
82
Chapter 8 Flow in Pipes
821C In fully developed laminar flow in a circular pipe, the head loss is given by
g
V
D
L
D g
V
D
L
D V g
V
D
L
g
V
D
L
f h
L
2
64
2 /
64
2 Re
64
2
2 2 2
?
?
= = = =
The average velocity can be expressed in terms of the flow rate as
4 /
2
D A
c
p
V V
& &
= = V . Substituting,
4 2 2 2 2
128
2
4 64
4 / 2
64
D g
L
D g
L
D D g
L
D
h
L
p
?
p
?
p
? V V V
& & &
= =
?
?
?
?
?
?
?
?
=
Therefore, at constant flow rate and pipe length, the head loss is inversely proportional to the 4
th
power of
diameter, and thus reducing the pipe diameter by half will increase the head loss by a factor of 16 .
822C In turbulent flow, it is the turbulent eddies due to enhanced mixing that cause the friction factor to
be larger.
823C Turbulent viscosity µ
t
is caused by turbulent eddies, and it accounts for momentum transport by
turbulent eddies. It is expressed as
y
u
v u
t t
?
?
= ' '  = µ ? t where u is the mean value of velocity in the
flow direction and and are the fluctuating components of velocity. u ' u '
824C We compare the dimensions of the two sides of the equation
5
2
0826 . 0
D
fL h
L
V
&
= ,
[] [ ] [ ][][
5
2
1 3
0826 . 0
 
· · · = L T L L L ],
and the dimension of the constant is
[] [ ]
2 1
T L 0826 . 0

=
Therefore, the constant 0.0826 is NOT dimensionless. This is not a dimensionally homogeneous, and it
cannot be used in any consistent set of units..
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
83
Page 4
Chapter 8 Flow in Pipes
Chapter 8
FLOW IN PIPES
Laminar and Turbulent Flow
81C Liquids are usually transported in circular pipes because pipes with a circular crosssection can
withstand large pressure differences between the inside and the outside without undergoing any significant
distortion.
82C Reynolds number is the ratio of the inertial forces to viscous forces, and it serves as a criteria for
determining the flow regime. At large Reynolds numbers, for example, the flow is turbulent since the
inertia forces are large relative to the viscous forces, and thus the viscous forces cannot prevent the random
and rapid fluctuations of the fluid. It is defined as follows:
(a) For flow in a circular tube of inner diameter D:
?
VD
= Re
(b) For flow in a rectangular duct of crosssection a × b:
?
h
VD
= Re
where D
A
p
ab
ab
ab
ab
h
c
==
+
=
+
4 4
2
2
() ( )
is the hydraulic diameter.
81
83C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
noting that Re = VD/ ?, the Reynolds number is higher for motion in water for the same diameter and speed.
a
D
b
84C Reynolds number for flow in a circular tube of diameter D is expressed as
?
VD
= Re where
?
µ
?
?p p ? ?
= = = = = and
4
) 4 / (
2 2
D
m
D
m
A
m
V
c
avg
& & &
V
Substituting,
m, V
µ p ? µ ?p ? D
m
D
D m VD & & 4
) / (
4
Re
2
= = =
85C Engine oil requires a larger pump because of its much larger density.
86C The generally accepted value of the Reynolds number above which the flow in a smooth pipe is
turbulent is 4000.
87C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
for the same diameter and speed, the Reynolds number will be higher for water flow, and thus the flow is
more likely to be turbulent for water.
88C For flow through noncircular tubes, the Reynolds number and the friction factor are based on the
hydraulic diameter D
h
defined as
p
A
D
c
h
4
= where A
c
is the crosssectional area of the tube and p is its
perimeter. The hydraulic diameter is defined such that it reduces to ordinary diameter D for circular tubes
since D
D
D
p
A
D
c
h
= = =
p
p 4 / 4 4
2
.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
Chapter 8 Flow in Pipes
89C The region from the tube inlet to the point at which the boundary layer merges at the centerline is
called the hydrodynamic entrance region, and the length of this region is called hydrodynamic entry length.
The entry length is much longer in laminar flow than it is in turbulent flow. But at very low Reynolds
numbers, L
h
is very small (L
h
= 1.2D at Re = 20).
810C The wall shear stress t
w
is highest at the tube inlet where the thickness of the boundary layer is zero,
and decreases gradually to the fully developed value. The same is true for turbulent flow.
811C In turbulent flow, the tubes with rough surfaces have much higher friction factors than the tubes
with smooth surfaces, and thus much larger pressure drop. In the case of laminar flow, the effect of surface
roughness on the friction factor and pressure drop is negligible.
Fully Developed Flow in Pipes
812C The wall shear stress t
w
remains constant along the flow direction in the fully developed region in
both laminar and turbulent flow.
813C The fluid viscosity is responsible for the development of the velocity boundary layer.
814C In the fully developed region of flow in a circular pipe, the velocity profile will NOT change in the
flow direction.
815C The friction factor for flow in a tube is proportional to the pressure loss. Since the pressure loss
along the flow is directly related to the power requirements of the pump to maintain flow, the friction factor
is also proportional to the power requirements to overcome friction. The applicable relations are
?
?
L
L
P m
W
V
D
L
f P
?
= = ?
&
&
pump
2
and
2
816C The shear stress at the center of a circular tube during fully developed laminar flow is zero since the
shear stress is proportional to the velocity gradient, which is zero at the tube center.
817C Yes, the shear stress at the surface of a tube during fully developed turbulent flow is maximum since
the shear stress is proportional to the velocity gradient, which is maximum at the tube surface.
818C In fully developed flow in a circular pipe with negligible entrance effects, if the length of the pipe is
doubled, the head loss will also double (the head loss is proportional to pipe length).
819C Yes, the volume flow rate in a circular pipe with laminar flow can be determined by measuring the
velocity at the centerline in the fully developed region, multiplying it by the crosssectional area, and
dividing the result by 2 since .
c c
A V A V ) 2 / (
max avg
= = V
&
820C No, the average velocity in a circular pipe in fully developed laminar flow cannot be determined by
simply measuring the velocity at R/2 (midway between the wall surface and the centerline). The average
velocity is V
max
/2, but the velocity at R/2 is
4
3
1 ) 2 / (
max
2 /
2
2
max
V
R
r
V R V
R r
=
?
?
?
?
?
?
?
?
 =
=
, which is much larger than V
max
/2.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
82
Chapter 8 Flow in Pipes
821C In fully developed laminar flow in a circular pipe, the head loss is given by
g
V
D
L
D g
V
D
L
D V g
V
D
L
g
V
D
L
f h
L
2
64
2 /
64
2 Re
64
2
2 2 2
?
?
= = = =
The average velocity can be expressed in terms of the flow rate as
4 /
2
D A
c
p
V V
& &
= = V . Substituting,
4 2 2 2 2
128
2
4 64
4 / 2
64
D g
L
D g
L
D D g
L
D
h
L
p
?
p
?
p
? V V V
& & &
= =
?
?
?
?
?
?
?
?
=
Therefore, at constant flow rate and pipe length, the head loss is inversely proportional to the 4
th
power of
diameter, and thus reducing the pipe diameter by half will increase the head loss by a factor of 16 .
822C In turbulent flow, it is the turbulent eddies due to enhanced mixing that cause the friction factor to
be larger.
823C Turbulent viscosity µ
t
is caused by turbulent eddies, and it accounts for momentum transport by
turbulent eddies. It is expressed as
y
u
v u
t t
?
?
= ' '  = µ ? t where u is the mean value of velocity in the
flow direction and and are the fluctuating components of velocity. u ' u '
824C We compare the dimensions of the two sides of the equation
5
2
0826 . 0
D
fL h
L
V
&
= ,
[] [ ] [ ][][
5
2
1 3
0826 . 0
 
· · · = L T L L L ],
and the dimension of the constant is
[] [ ]
2 1
T L 0826 . 0

=
Therefore, the constant 0.0826 is NOT dimensionless. This is not a dimensionally homogeneous, and it
cannot be used in any consistent set of units..
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
are using it without permission.
83
Chapter 8 Flow in Pipes
825C In fully developed laminar flow in a circular pipe, the pressure loss and the head loss are given by
2
32
D
LV
P
L
µ
= ? and
2
32
gD
LV
g
P
h
L
L
?
µ
?
=
?
=
When the flow rate and thus average velocity are held constant, the head loss becomes proportional to
viscosity. Therefore, the head loss will be reduced by half when the viscosity of the fluid is reduced by
half.
826C The head loss is related to pressure loss by g P h
L L
? / ? = . For a given fluid, the head loss can be
converted to pressure loss by multiplying the head loss by the acceleration of gravity and the density of the
fluid.
827C During laminar flow of air in a circular pipe with perfectly smooth surfaces, the friction factor will
NOT be zero because of the noslip boundary condition. But it will be minimum compared to flow in pipes
with rough surfaces.
828C At very large Reynolds numbers, the flow is fully rough and the friction factor is independent of the
Reynolds number. This is because the thickness of laminar sublayer decreases with increasing Reynolds
number, and it be comes so thin that the surface roughness protrudes into the flow. The viscous effects in
this case are produced in the main flow primarily by the protruding roughness elements, and the
contribution of the laminar sublayer becomes negligible.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
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Page 5
Chapter 8 Flow in Pipes
Chapter 8
FLOW IN PIPES
Laminar and Turbulent Flow
81C Liquids are usually transported in circular pipes because pipes with a circular crosssection can
withstand large pressure differences between the inside and the outside without undergoing any significant
distortion.
82C Reynolds number is the ratio of the inertial forces to viscous forces, and it serves as a criteria for
determining the flow regime. At large Reynolds numbers, for example, the flow is turbulent since the
inertia forces are large relative to the viscous forces, and thus the viscous forces cannot prevent the random
and rapid fluctuations of the fluid. It is defined as follows:
(a) For flow in a circular tube of inner diameter D:
?
VD
= Re
(b) For flow in a rectangular duct of crosssection a × b:
?
h
VD
= Re
where D
A
p
ab
ab
ab
ab
h
c
==
+
=
+
4 4
2
2
() ( )
is the hydraulic diameter.
81
83C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
noting that Re = VD/ ?, the Reynolds number is higher for motion in water for the same diameter and speed.
a
D
b
84C Reynolds number for flow in a circular tube of diameter D is expressed as
?
VD
= Re where
?
µ
?
?p p ? ?
= = = = = and
4
) 4 / (
2 2
D
m
D
m
A
m
V
c
avg
& & &
V
Substituting,
m, V
µ p ? µ ?p ? D
m
D
D m VD & & 4
) / (
4
Re
2
= = =
85C Engine oil requires a larger pump because of its much larger density.
86C The generally accepted value of the Reynolds number above which the flow in a smooth pipe is
turbulent is 4000.
87C Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water
than for air (at 25 °C, ?
air
= 1.562 ×10
5
m
2
/s and ?
water
= µ/ ? = 0.891 ×10
3
/997 = 8.9 ×10
7
m
2
/s). Therefore,
for the same diameter and speed, the Reynolds number will be higher for water flow, and thus the flow is
more likely to be turbulent for water.
88C For flow through noncircular tubes, the Reynolds number and the friction factor are based on the
hydraulic diameter D
h
defined as
p
A
D
c
h
4
= where A
c
is the crosssectional area of the tube and p is its
perimeter. The hydraulic diameter is defined such that it reduces to ordinary diameter D for circular tubes
since D
D
D
p
A
D
c
h
= = =
p
p 4 / 4 4
2
.
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
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Chapter 8 Flow in Pipes
89C The region from the tube inlet to the point at which the boundary layer merges at the centerline is
called the hydrodynamic entrance region, and the length of this region is called hydrodynamic entry length.
The entry length is much longer in laminar flow than it is in turbulent flow. But at very low Reynolds
numbers, L
h
is very small (L
h
= 1.2D at Re = 20).
810C The wall shear stress t
w
is highest at the tube inlet where the thickness of the boundary layer is zero,
and decreases gradually to the fully developed value. The same is true for turbulent flow.
811C In turbulent flow, the tubes with rough surfaces have much higher friction factors than the tubes
with smooth surfaces, and thus much larger pressure drop. In the case of laminar flow, the effect of surface
roughness on the friction factor and pressure drop is negligible.
Fully Developed Flow in Pipes
812C The wall shear stress t
w
remains constant along the flow direction in the fully developed region in
both laminar and turbulent flow.
813C The fluid viscosity is responsible for the development of the velocity boundary layer.
814C In the fully developed region of flow in a circular pipe, the velocity profile will NOT change in the
flow direction.
815C The friction factor for flow in a tube is proportional to the pressure loss. Since the pressure loss
along the flow is directly related to the power requirements of the pump to maintain flow, the friction factor
is also proportional to the power requirements to overcome friction. The applicable relations are
?
?
L
L
P m
W
V
D
L
f P
?
= = ?
&
&
pump
2
and
2
816C The shear stress at the center of a circular tube during fully developed laminar flow is zero since the
shear stress is proportional to the velocity gradient, which is zero at the tube center.
817C Yes, the shear stress at the surface of a tube during fully developed turbulent flow is maximum since
the shear stress is proportional to the velocity gradient, which is maximum at the tube surface.
818C In fully developed flow in a circular pipe with negligible entrance effects, if the length of the pipe is
doubled, the head loss will also double (the head loss is proportional to pipe length).
819C Yes, the volume flow rate in a circular pipe with laminar flow can be determined by measuring the
velocity at the centerline in the fully developed region, multiplying it by the crosssectional area, and
dividing the result by 2 since .
c c
A V A V ) 2 / (
max avg
= = V
&
820C No, the average velocity in a circular pipe in fully developed laminar flow cannot be determined by
simply measuring the velocity at R/2 (midway between the wall surface and the centerline). The average
velocity is V
max
/2, but the velocity at R/2 is
4
3
1 ) 2 / (
max
2 /
2
2
max
V
R
r
V R V
R r
=
?
?
?
?
?
?
?
?
 =
=
, which is much larger than V
max
/2.
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82
Chapter 8 Flow in Pipes
821C In fully developed laminar flow in a circular pipe, the head loss is given by
g
V
D
L
D g
V
D
L
D V g
V
D
L
g
V
D
L
f h
L
2
64
2 /
64
2 Re
64
2
2 2 2
?
?
= = = =
The average velocity can be expressed in terms of the flow rate as
4 /
2
D A
c
p
V V
& &
= = V . Substituting,
4 2 2 2 2
128
2
4 64
4 / 2
64
D g
L
D g
L
D D g
L
D
h
L
p
?
p
?
p
? V V V
& & &
= =
?
?
?
?
?
?
?
?
=
Therefore, at constant flow rate and pipe length, the head loss is inversely proportional to the 4
th
power of
diameter, and thus reducing the pipe diameter by half will increase the head loss by a factor of 16 .
822C In turbulent flow, it is the turbulent eddies due to enhanced mixing that cause the friction factor to
be larger.
823C Turbulent viscosity µ
t
is caused by turbulent eddies, and it accounts for momentum transport by
turbulent eddies. It is expressed as
y
u
v u
t t
?
?
= ' '  = µ ? t where u is the mean value of velocity in the
flow direction and and are the fluctuating components of velocity. u ' u '
824C We compare the dimensions of the two sides of the equation
5
2
0826 . 0
D
fL h
L
V
&
= ,
[] [ ] [ ][][
5
2
1 3
0826 . 0
 
· · · = L T L L L ],
and the dimension of the constant is
[] [ ]
2 1
T L 0826 . 0

=
Therefore, the constant 0.0826 is NOT dimensionless. This is not a dimensionally homogeneous, and it
cannot be used in any consistent set of units..
PROPRIETARY MATERIAL. © 2006 The McGrawHill Companies, Inc. Limited distribution
permitted only to teachers and educators for course preparation. If you are a student using this Manual, you
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83
Chapter 8 Flow in Pipes
825C In fully developed laminar flow in a circular pipe, the pressure loss and the head loss are given by
2
32
D
LV
P
L
µ
= ? and
2
32
gD
LV
g
P
h
L
L
?
µ
?
=
?
=
When the flow rate and thus average velocity are held constant, the head loss becomes proportional to
viscosity. Therefore, the head loss will be reduced by half when the viscosity of the fluid is reduced by
half.
826C The head loss is related to pressure loss by g P h
L L
? / ? = . For a given fluid, the head loss can be
converted to pressure loss by multiplying the head loss by the acceleration of gravity and the density of the
fluid.
827C During laminar flow of air in a circular pipe with perfectly smooth surfaces, the friction factor will
NOT be zero because of the noslip boundary condition. But it will be minimum compared to flow in pipes
with rough surfaces.
828C At very large Reynolds numbers, the flow is fully rough and the friction factor is independent of the
Reynolds number. This is because the thickness of laminar sublayer decreases with increasing Reynolds
number, and it be comes so thin that the surface roughness protrudes into the flow. The viscous effects in
this case are produced in the main flow primarily by the protruding roughness elements, and the
contribution of the laminar sublayer becomes negligible.
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84
Chapter 8 Flow in Pipes
829E The pressure readings across a pipe are given. The flow rates are to be determined for 3 different
orientations of horizontal, uphill, and downhill flow. v
Assumptions 1 The flow is steady and incompressible. 2 The entrance effects are negligible, and thus the
flow is fully developed. 3 The flow is laminar (to be verified). 4 The pipe involves no components such as
bends, valves, and connectors. 5 The piping section involves no work devices such as pumps and turbines.
Properties The density and dynamic viscosity of oil are given to be ? = 56.8 lbm/ft
3
and µ = 0.0278
lbm/ft ·s, respectively.
Analysis The pressure drop across the pipe and the crosssectional area of the pipe are
2 2 2
2 1
ft 001364 . 0 4 / ft) 12 / 5 . 0 ( 4 /
psi 106 14 120
= = =
=  =  = ?
p pD A
P P P
c
Oil
D = 0.5 in
L = 120 ft
(a) The flow rate for all three cases can be determined from
L
D gL P
µ
p ? ?
128
) sin (
4
 ?
= V
&
where ? is the angle the pipe makes with the horizontal. For the horizontal case, ? = 0 and thus sin ? = 0.
Therefore,
/s ft 0.0109
3
=
?
?
?
?
?
?
?
?
·
?
?
?
?
?
?
?
?
·
=
?
=
lbf 1
ft/s lbm 2 . 32
psi 1
lbf/ft 144
ft) s)(120 lbm/ft 0278 . 0 ( 128
ft) (0.5/12 psi) 106 (
128
2 2 4 4
horiz
p
µ
p
L
D P
V
&
(b) For uphill flow with an inclination of 20 °, we have ? = +20 °, and
psi 2 . 16
ft/s lbm 2 . 32
lbf 1
lbf/ft 144
psi 1
20 sin ) ft 120 )( ft/s 2 . 32 )( lbm/ft 8 . 56 ( sin
2 2
2 3
= ?
?
?
?
?
?
·
?
?
?
?
?
?
° = ? ?gL
/s ft 0.00923
3
=
?
?
?
?
?
?
?
?
·
?
?
?
?
?
?
?
?
·

=
=
 ?
=
lbf 1
ft/s lbm 2 . 32
psi 1
lbf/ft 144
ft) s)(120 lbm/ft 0278 . 0 ( 128
ft) (0.5/12 psi) 2 . 16 106 (
128
) sin (
2 2 4
4
uphill
p
µ
p ? ?
L
D gL P
V
&
20°
(c) For downhill flow with an inclination of 20 °, we have ? = 20 °, and
/s ft 0.0126
3
=
?
?
?
?
?
?
?
?
·
?
?
?
?
?
?
?
?
·
 
=
 ?
=
lbf 1
ft/s lbm 2 . 32
psi 1
lbf/ft 144
ft) s)(120 lbm/ft 0278 . 0 ( 128
ft) (0.5/12 psi] ) 2 . 16 ( 106 [
128
) sin (
2 2 4
4
downhill
p
µ
p ? ?
L
D gL P
V
&
20°
The flow rate is the highest for downhill flow case, as expected. The average fluid velocity and the
Reynolds number in this case are
787
s lbm/ft 0278 . 0
ft) 12 ft/s)(0.5/ 24 . 9 )( lbm/ft 8 . 56 (
Re
ft/s 24 . 9
ft 0.001364
/s ft 0126 . 0
3
2
3
=
·
= =
= = =
µ
?VD
A
V
c
V
&
which is less than 2300. Therefore, the flow is laminar for all three cases, and the analysis above is valid.
Discussion Note that the flow is driven by the combined effect of pressure difference and gravity. As can
be seen from the calculated rates above, gravity opposes uphill flow, but helps downhill flow. Gravity has
no effect on the flow rate in the horizontal case. Downhill flow can occur even in the absence of an applied
pressure difference.
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are using it without permission.
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