Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
CHAPTER FIVE
VISCOUS FLOW IN PIPE
5.1 General characteristics of pipe flow
Before we apply the governing equation to pipe flow we will discuss some of
the basic concepts of pipe flow, we will assume that the conduit is round .
For all flows involved in this chapter we assume that the pipe is completely
filled with the fluid being transported.
5.1.1 Laminar or Turbulent Flow
The flow of fluid in pipe may be laminar flow
or it may be turbulent flow. if water runs
through a pipe of diameter (D) with average
velocity (V), the following characteristics are
observed by injecting naturally buoyant dye
as shown.
for small enough flow rate the dye streak will
remain as well defined line as it flows along
(laminar flow), for somewhat large the dye streak fluctuates in time and space
(transitional flow).
For large enough flow rate the dye streak almost immediately becomes
blurred and spread across the entire pipe in random fashion (Turbulent flow).
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 1 of 11
Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
The next curves represent the x-component of velocity as a function of time at
a point (A) in the flow.
For pipe flow the most important dimensionless parameter is Reynolds
number.
Reynolds Number (Re): is the ratio of the inertia effect to viscous effect in the
flow.
𝑅𝑒 =
𝜌𝑉𝐷
𝜇
V : average velocity in the pipe.
 : density of fluid
 : viscosity of fluid
In general the flow in round pipe
Re < 2100
Laminar for
Transitional for
2100 < Re < 4000
Turbulent for
Re > 4000
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 2 of 11
Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Example:
Water at a temperature of 20 oC flows through pipe
of diameter D = 2 cm .
Determine: (a) the minimum time taken to fill a
glass of volume 0.5 l , if the flow in pipe is laminar.
(b) the maximum time taken to fill the glass if the
flow is turbulent.
Solution:
(a) if the flow in pipe is to remain laminar the
minimum time to fill the glass will occur if the Re
is the maximum allowed , Re = 2100
From the table B.1 for water at 20 oC
𝜌 = 998.2 𝑘𝑔⁄𝑚3
𝜇 = 1.002 × 10−3 𝑁. 𝑠⁄𝑚2
𝜌𝑉𝐷
𝑅𝑒 𝜇 2100 × 1.002 × 10−3
𝑅𝑒 =
= 2100 ⇒ 𝑉 =
=
𝜇
𝜌𝐷
998.2 × 0.02
V = 0.1 m/s
𝑄 = 𝐴𝑉 =
𝜋 2
𝜋
𝐷 𝑉 = (0.02)2 (0.1) = 3.14 × 10−5 𝑚3 ⁄𝑠
4
4
the time
∀
0.5 × 10−3
𝑡= =
= 15.9 𝑠𝑒𝑐
𝑄 3.14 × 10−5
(b) if the flow in pipe is turbulent the maximum time occur at
Re = 4000 ⇒
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 3 of 11
Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
𝑅𝑒 𝜇 4000 × 1.002 × 10−3
𝑉=
=
= 0.2 𝑚/𝑠
𝜌𝐷
998.2 × 0.02
𝑄 = 𝐴𝑉 =
𝜋
4
𝜋
𝐷2 𝑉 = (0.02)2 (0.2) = 6.28 × 10−5 𝑚3 ⁄𝑠
4
∀
0.5 × 10−3
𝑡= =
= 7.96 𝑠𝑒𝑐
𝑄 6.28 × 10−5
5.1.2 Entrance region and fully developed Flow
Any fluid in pipe has to enter the pipe at some location the region of flow near
where the fluid enters the pipe is termed the entrance region as in figure.
The fluid typically enters the pipe with nearly uniform velocity profile at
section (1) , as the fluid moves through the pipe , viscous effects cause it to
stick to the pipe wall (the no slip condition). thus a boundary layer is produced
along the fluid reaches the end of the entrance length at section (2), beyond
which the velocity profile dose not vary with x.
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 4 of 11
Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
5.1.3 Energy Equation
For one-dimensional, incompressible, steady flow with friction and shaft
work. The energy equation is,
𝑝 𝑉2
𝑝 𝑉2
+ 𝑔𝑍) = ( +
+ 𝑔𝑍) + 𝑊𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡 + 𝑙𝑜𝑠𝑠𝑒𝑠
( +
𝜌 2
𝜌
2
1
2
where
𝑊𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡
𝑊̇𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡
=
𝑚̇
(𝑁. 𝑚⁄𝑘𝑔 = 𝑚2 ⁄𝑠 2 )
Dividing by gravity (g)
𝑝 𝑉2
𝑝 𝑉2
+ 𝑍) = ( +
+ 𝑍) + ℎ𝑠 + ℎ𝐿
( +
𝛾 2𝑔
𝛾
2𝑔
1
2
ℎ𝑠 =
ℎ𝐿 =
𝑊𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡 𝑊̇𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡
𝑊̇𝑠ℎ𝑎𝑓𝑡 𝑛𝑒𝑡
=
=
𝑔
𝑚̇ 𝑔
𝛾𝑄
𝑙𝑜𝑠𝑠𝑒𝑠
= 𝑒𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡
𝑔
(𝑚)
( 𝑁. 𝑚⁄𝑁 ) = (𝑚)
5.2 Dimensional analysis of pipe flow:
It is necessary to determine the head loss ( hL ) - (pressure drop) that occurs
in a pipe flow. The overall head loss for the pipe system consists of the head
loss due to viscous effects in the straight pipes, termed the Major loss, and
denoted ( hL major ) and the head loss in the various pipe components (valves,
elbows, … etc.), termed the, Minor loss and denoted ( hL minor ).
hL = hL major + hL minor
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 5 of 11
Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
The head loss designations of “major” and “minor” do not necessarily reflect
the relative importance of each type of loss. For a pipe system that contains
many components and a relatively short length of pipe, the minor loss may
actually be larger than the major loss.
5.2.1 Major losses
For any fully developed, steady, incompressible pipe flow - whether the pipe
is horizontal or on a hill, the Darcy-Weisbach is used to determine the major
head loss ( hL major ),
𝑙 𝑉2
ℎ𝐿 𝑚𝑎𝑗𝑜𝑟 = 𝑓
𝐷 2𝑔
where
V
is average velocity
l
is pipe length
D is pipe diameter
f
is friction factor ,
The friction factor f is dependent on
the behavior of the flow.
for Laminar flow, 𝑓 = 64⁄𝑅𝑒
for Turbulent flow , f is determined from the Colebrook formula,
𝜀 ⁄𝐷
2.51
= −2.0 log (
+
)
3.7 𝑅𝑒 √𝑓
√𝑓
1
𝜀 ⁄𝐷 ∶ 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑅𝑜𝑢𝑔ℎ𝑛𝑒𝑠𝑠
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 6 of 11
‫‪Course Notes in Fluid Mechanics‬‬
‫‪University of Benghazi‬‬
‫‪Lecturer: Dr. Nabil Elsharif‬‬
‫‪Faculty of Engineering‬‬
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
‫‪The Colebrook equation is represented in chart , which is called Moody chart.‬‬
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
‫‪Page 7 of 11‬‬
‫‪Chapter (5) - Viscous Flow in Pipes‬‬
Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Example:
Air under standard conditions flows through a 4.0-mm diameter drawn tubing
with an average velocity of V = 50 m/s. For such conditions the flow would
normally be turbulent. However, if precautions are taken to eliminate
disturbances to the flow (the entrance to the tube is very smooth, the air is
dust free, the tube does not vibrate, etc.), it may be possible to maintain
laminar flow. (a) Determine the head loss in a 0.1-m section of the tube if the
flow is laminar.
(b) Repeat the calculations if the flow is turbulent.
(c) Calculations the pressure drop in the tube
Solution:
for standard temperature and pressure conditions the density and viscosity
of air are
𝜌 = 1.23 𝑘𝑔⁄𝑚3
𝜇 = 1.79 × 10−5 𝑁. 𝑠⁄𝑚2
𝑅𝑒 =
𝜌 𝑉 𝐷 1.23 × 50 × 0.004
=
= 13,700
𝜇
1.79 × 10−5
which would normally indicate turbulent flow.
(a) If the flow were laminar, then 𝑓 = 64⁄𝑅𝑒 = 64⁄13700 = 0.00467
and the pressure drop in a 0.1-m-long horizontal section of the pipe would be
(50)2
𝑙 𝑉2
0.1
ℎ𝐿 = 𝑓
= (0.00467) ×
×
= 14.8 𝑚
𝐷 2𝑔
0.004 2 × 9.81
(b) If the flow is turbulent
from table (8.1) 𝜀 = 0.0015 𝑚𝑚, so that
𝜀
0.0015
=
= 0.000375
𝐷
4
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 8 of 11
Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
From Moody chart at
𝑅𝑒 = 1.37 × 104 𝑎𝑛𝑑
f = 0.028
𝜀
𝐷
= 0.000375
(50)2
𝑙 𝑉2
0.1
(0.028)
ℎ𝐿 = 𝑓
=
×
×
= 89.1 𝑚
𝐷 2𝑔
0.004 2 × 9.81
(c) pressure drop in the tube
from energy equation
𝑝 𝑉2
𝑝 𝑉2
+ 𝑍) = ( +
+ 𝑍) + ℎ𝑠 + ℎ𝐿
( +
𝛾 2𝑔
𝛾
2𝑔
1
2
𝑍1 = 𝑍2
,
𝑉1 = 𝑉2
𝑝1− 𝑝2
∆𝑝
=
= ℎ𝐿 ⇒
𝛾
𝛾
,
hs = 0
∆𝑝 = 𝜌 𝑔 ℎ𝐿
for Laminar flow
∆𝑝 = 1.23 × 9.81 × 14.8 = 0.179 𝑘𝑃𝑎
for Turbulent flow
∆𝑝 = 1.23 × 9.81 × 89.1 = 1.076 𝑘𝑃𝑎
5.2.2 Minor losses
this losses are due to the components (valves, elbows, bends, tees … etc.)
added to the pipe system. The most common method used to determine these
head losses or pressure drops is to specify the loss coefficient, KL which is
defined as
ℎ𝐿 𝑚𝑖𝑛𝑜𝑟
∆𝑝
𝑉2
𝐾𝐿 = 2
=
⇒ ℎ𝐿 𝑚𝑖𝑛𝑜𝑟 = 𝐾𝐿
1 2
(𝑉 /2𝑔)
2𝑔
(2 𝜌𝑉 )
The actual value of KL is strongly dependent on the geometry of the
component considered. It may also be dependent on the fluid properties.
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 9 of 11
‫‪Course Notes in Fluid Mechanics‬‬
‫‪University of Benghazi‬‬
‫‪Lecturer: Dr. Nabil Elsharif‬‬
‫‪Faculty of Engineering‬‬
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
‫‪Typical values of KL for such components are given in Table 8.2.‬‬
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
‫‪Page 10 of 11‬‬
‫‪Chapter (5) - Viscous Flow in Pipes‬‬
Course Notes in Fluid Mechanics
University of Benghazi
Lecturer: Dr. Nabil Elsharif
Faculty of Engineering
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
•
The energy equation for incompressible, steady flow between two
locations is given by
5.2.3 Noncircular Conduits
For noncircular ducts the previous correlation can be used by introducing the
hydraulic diameter ( Dh ) where
𝐷ℎ =
4𝐴
𝑃
A : cross sectional area of the duct
P : the wetted perimeter of the duct
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
Chapter (5) - Viscous Flow in Pipes
Page 11 of 11