Viscous Flow in Pipes
CEE 331 Fluid Mechanics
March 14, 2016
Monroe L. Weber-Shirk
School of Civil and
Environmental Engineering
Types of Engineering Problems
 How
big does the pipe have to be to carry a
flow of x m3/s?
 What will the pressure in the water
distribution system be when a fire hydrant is
open?
 Can we increase the flow in this old pipe by
adding a smooth liner?
Viscous Flow in Pipes: Overview
 Boundary
Layer Development
 Turbulence
 Velocity Distributions
 Energy Losses
 Major
 Minor
 Solution Techniques
Laminar and Turbulent Flows
 Reynolds
apparatus
r VD inertia
Re =
=
damping
m
Transition at Re of 2000
Boundary layer growth:
Transition length
What does the water near the pipeline wall experience?
Drag or shear
_________________________
Why does the water in the center of the pipeline speed
Conservation of mass
up? _________________________
Non-Uniform Flow
v
v
v
Entrance Region Length
le
= f (Re)
D
le
= 0.06 Re
D
le
1/ 6
= 4.4 (Re)
D
100
10
laminar
Re
0
00
00
00
10
10
00
00
00
0
00
00
10
00
00
10
0
00
10
00
10
0
1
10
l e /D
10
Distance
for
velocity
profile to
develop
turbulent
Velocity Distributions
 Turbulence
causes transfer of momentum
from center of pipe to fluid closer to the pipe
wall.
 Mixing of fluid (transfer of momentum)
causes the central region of the pipe to have
relatively constant
_______velocity (compared to
laminar flow)
 Close to the pipe wall eddies are smaller (size
proportional to distance to the boundary)
Log Law for Turbulent, Established
Flow, Velocity Profiles
u
yu
= 2.5ln * + 5.0 Dimensional analysis and measurements
u*
n
Turbulence produced by shear!
u* 
0
Shear velocity u*  uI Velocity of large eddies

g hl d
4l
ghl d
u* =
4l
t0=
rough
Force balance
smooth
y
u* = V f
u/umax
Pipe Flow: The Problem
 We
have the control volume energy
equation for pipe flow
 We need to be able to predict the head loss
term.
 We will use the results we obtained using
dimensional analysis
Viscous Flow: Dimensional
Analysis
 Remember
dimensional analysis?
D
æe
ö
Cp = f
, Re
èD
ø
l
 Two
 2p
r VD
and C p 
Where Re =
V 2
m
important parameters!
 Re
- Laminar or Turbulent
 e/D - Rough or Smooth
 Flow
geometry
 internal
_______________________________
in a bounded region (pipes, rivers): find Cp
 external _______________________________
flow around an immersed object : find Cd
Pipe Flow Energy Losses
 D
e

f   C p   f  , R 

D 
L
 2p
Cp 
V 2
2 gh D
f  2l
V L
LV2
hl  f
D 2g
Dimensional Analysis
r ghl = - Dp - r g Dz
r ghl = - Dp
Cp 
2 ghl
More general
V2
Assume horizontal flow
Always true (laminar or turbulent)
Darcy-Weisbach equation
Friction Factor : Major losses
 Laminar
flow
 Turbulent (Smooth, Transition, Rough)
 Colebrook Formula
 Moody diagram
 Swamee-Jain
Laminar Flow Friction Factor
r gD 2 hl
V=
32 m L
hl 
32LV
gD 2
LV2
hl  f
D 2g
32LV
LV2
f
2
D 2g
gD
64  64
f

VD R
Hagen-Poiseuille
hl µ V
Darcy-Weisbach
f independent of roughness!
Slope of ___
-1 on log-log plot
LV2
hf = f
D 2g
Turbulent Flow:
Smooth, Rough, Transition
Hydraulically smooth
pipe law (von Karman,
1930)
 Rough pipe law (von
Karman, 1930)
 Transition function for
both smooth and rough
pipe laws (Colebrook)

æRe f ö
1
= 2 log ç
÷
f
è 2.51 ø
1
æ3.7 D ö
= 2 log
è e ø
f
1
2.51 ö
æe D
= - 2 log
+
è
3.7
f
Re f ø
(used to draw the Moody diagram)
Moody Diagram
0.10
0.08
 D
f  Cp 
l  0.06

0.05
0.04
0.03
friction factor
0.05
0.02
0.015
0.04
0.01
0.008
0.006
0.004
0.03
laminar
0.002
0.02
0.001
0.0008
0.0004
0.0002
0.0001
0.00005
0.01
1E+03
smooth
1E+04
1E+05
1E+06
Re
1E+07
1E+08
e
D
Swamee-Jain


1976
limitations




e/D < 2 x 10-2
Re >3 x 103
less than 3% deviation
from results obtained
with Moody diagram
easy to program for
computer or calculator
use
f=
0.25
2
5.74 öù
é æe
log
+ 0.9 ú
ê
è
3.7 D Re øû
ë
Q = - 0.965D 2
no f
æ
gDhf ç e
1.784n
ln ç
+
L
ç3.7 D D gDhf
è
L
é 1.25 æLQ ö
D = 0.66 êe ç
÷
è
gh
f ø
ê
ë
2
4.75
ö
÷
÷
÷
ø
5.2 0.04
æL ö
+n Q ç ÷
è ghf ø
9.4
ù
ú
ú
û
Each equation has two terms. Why?
Pipe roughness
pipe material
glass, drawn brass, copper
commercial steel or wrought iron
asphalted cast iron
galvanized iron
cast iron
concrete
rivet steel
corrugated metal
PVC
pipe roughness e (mm)
0.0015
0.045
e
0.12
d Must be
0.15 dimensionless!
0.26
0.18-0.6
0.9-9.0
45
0.12
Solution Techniques
find
head loss given (D, type of pipe, Q)
2
0.25
8
LQ
4Q
f=
2
hf = f 2
Re 
5
e
5.74
é
ù
æ
ö
p
g
D
D
log
+ 0.9 ú
ê
ë è3.7 D Re øû
find flow rate given (head, D, L, type of pipe)
æ
ö
gDhf ç e
1.784n ÷
2
Q = - 0.965D
ln ç
+
÷
L
3.7
D
gDh
f ÷
ç
D
è
L ø
find pipe size given (head, type of pipe,L, Q)
é 1.25 æLQ ö
D = 0.66 êe ç
÷
è
gh
f ø
ê
ë
2
4.75
5.2 0.04
æL ö
+n Q ç ÷
è ghf ø
9.4
ù
ú
ú
û
Example: Find a pipe diameter

The marine pipeline for the Lake Source Cooling project
will be 3.1 km in length, carry a maximum flow of 2 m3/s,
and can withstand a maximum pressure differential
between the inside and outside of the pipe of 28 kPa. The
pipe roughness is 2 mm. What diameter pipe should be
used?
Minor Losses
 We
previously obtained losses through an
expansion using conservation of energy,
momentum, and mass
 Most minor losses can not be obtained
analytically, so they must be measured
 Minor losses are often expressed as a loss
V2
coefficient, K, times the velocity head. hl = K 2 g
High Re
C p = f (geometry, Re) C p 
 2p
V 2
Cp 
2 ghl
V2
hl  C p
V2
2g
Venturi
hexpansion
2
V æ Ain ö
=
1ç
2g è
Aout ÷
ø
2
in
Sudden Contraction
EGL
Cc 
2
 1
 V2
hc  
 1 2
C
 2g
 c

HGL
Ac
A2
1
0.95
0.9
0.85
Cc 0.8
0.75
0.7
0.65
0.6
0
0.2
0.4
0.6
A2/A1
0.8
1
c
2
V1


vena contracta
Losses are reduced with a gradual contraction
Equation has same form as expansion equation!
V2
Entrance Losses
V2
he  K e
Losses can be
2g
reduced by
K e  1.0
accelerating the flow
Estimate based on
contraction equations!
gradually and
K e  0.5
eliminating the
vena contracta
K e  0.04
Head Loss in Bends
High pressure
 Head
loss is a function
of the ratio of the bend
radius to the pipe
diameter (R/D)
 Velocity distribution
D
returns to normal
several pipe diameters
hb
downstream
Kb varies from 0.6 - 0.9
R
Possible
separation
from wall
n
2
óV
p + r ô dn + g z = C
õ R
Low pressure
 Kb
V2
2g
Head Loss in Valves
Function of valve type and valve
position
 The complex flow path through
valves can result in high head loss
(of course, one of the purposes of a
valve is to create head loss when it
is not fully open)
 see table 8.2 (page 325 in text)

¥
What is the maximum value of Kv? ______
hv  K v
V2
2g
Solution Techniques
 Neglect
minor losses
 Equivalent pipe lengths
 Iterative Techniques
 Using
Swamee-Jain equations for D and Q
 Using Swamee-Jain equations for head loss
 Assume a friction factor
 Pipe
Network Software
Solution Technique 1:
Find D or Q
 Assume
all head loss is major head loss
 Calculate D or Q using Swamee-Jain
equations
8Q 2
hminor = K
 Calculate minor losses
gp 2 D 4
 Find new major losses by subtracting minor
losses from total head loss hf = hl - å hminor
Q = - 0.965D 2
æ
gDhf ç e
1.784n
ln ç
+
L
ç3.7 D D gDhf
è
L
ö
÷
÷
÷
ø
0.04
5.2
é 1.25 æLQ 2 ö4.75
ù
æ
ö
L
9.4
D = 0.66 êe ç
+n Q ç ÷ ú
÷
è ghf ø
è ghf ø ú
ê
ë
û
Solution Technique: Head Loss
 Can
be solved explicitly
hminor = å
Re 
4Q
D
V2
K
2g
f 
hminor
8Q 2
=
gp 2
0.25

 e
5.74 



log

 3.7 D Re 0.9 

hl = å hf + å hminor
2
å
K
D4
8 LQ 2
hf = f
gp 2 D 5
Solution Technique 2:
Find D or Q using Solver
 Iterative
technique
 Solve these equations
Re 
4Q
D
hminor  K
f =
0.25
2
5.74 öù
é
æ e
log
+
ê
è3.7 D Re0.9 øú
ë
û
8Q 2
g 2 D 4
hl = å hf + å hminor
8 LQ 2
hf = f
gp 2 D 5
Use goal seek or Solver to
find discharge that makes the
calculated head loss equal
the given head loss.
Spreadsheet
Solution Technique 3:
Find D or Q by assuming f
 The
friction factor doesn’t vary greatly
 If Q is known assume f is 0.02, if D is
1
æ3.7 D ö
known assume rough pipe law
= 2 log
è e ø
f
 Use Darcy Weisbach and minor loss
equations
 Solve for Q or D
 Calculate Re and e/D
 Find new f on Moody diagram
 Iterate
Example: Minor and Major
Losses

Find the maximum dependable flow between the
reservoirs for a water temperature range of 4ºC to
20ºC.
Water
25 m elevation difference in reservoir water levels
Reentrant pipes at reservoirs
Standard elbows
2500 m of 8” PVC pipe
Sudden contraction
1500 m of 6” PVC pipe
Gate valve wide open
Directions
Example (Continued)
 What
are the Reynolds numbers in the two
pipes? 90,000 & 125,000
 Where are we on the Moody Diagram?
 What is the effect of temperature?
 Why is the effect of temperature so small?
 What value of K would the valve have to
produce to reduce the discharge by 50%? 140
friction factor
0.1
0.05
0.04
0.03
0.02
0.015
0.01
0.008
0.006
0.004
laminar
0.01
1E+03
1E+04
1E+05
1E+06
Re
1E+07
1E+08
0.002
0.001
0.0008
0.0004
0.0002
0.0001
0.00005
smooth
Example (Continued)
Yes
the minor losses negligible?
5%
 Accuracy of head loss calculations?
 What happens if the roughness increases by
f goes from 0.02 to 0.035
a factor of 10?
 If you needed to increase the flow by 30%
what could you do? Increase small pipe diameter
 Were
0.1
0.05
0.04
0.03
friction factor
0.02
0.015
0.01
0.008
0.006
0.004
laminar
0.002
0.001
0.0008
0.0004
0.0002
0.0001
0.00005
0.01
1E+03
smooth
1E+04
1E+05
1E+06
Re
1E+07
1E+08
Pipe Flow Summary (1)
 Shear
increases _________
linearly with distance
from the center of the pipe (for both laminar
and turbulent flow)
 Laminar flow losses and velocity
distributions can be derived based on
momentum (Navier Stokes) and energy
conservation
 Turbulent flow losses and velocity
distributions require ___________
experimental results
Pipe Flow Summary (2)
Energy equation left us with the elusive head loss
term
 Dimensional analysis gave us the form of the head
loss term (pressure coefficient)
 Experiments gave us the relationship between the
pressure coefficient and the geometric parameters
and the Reynolds number (results summarized on
Moody diagram)

Pipe Flow Summary (3)
 Dimensionally
correct equations fit to the
empirical results can be incorporated into
computer or calculator solution techniques
 Minor losses are obtained from the pressure
coefficient based on the fact that the
pressure coefficient is _______
constant at high
Reynolds numbers
 Solutions for discharge or pipe diameter
often require iterative or computer solutions
Pipes are Everywhere!
Owner: City of
Hammond, IN
Project: Water Main
Relocation
Pipe Size: 54"
Pipes are Everywhere!
Drainage Pipes
Pipes
Pipes are Everywhere!
Water Mains
Pressure Coefficient for a Venturi
Meter
Cp
10
 2p
Cp 
V 2
1
1E+01
1E+02
1E+03
1E+04
1E+05
1E+06
Re
r Vl
Re =
m
0.1
0.05
0.04
0.03
0.02
0.015
friction factor
1E+00
0.01
0.008
0.006
0.004
laminar
0.002
0.001
0.0008
0.0004
0.0002
0.0001
0.00005
0.01
1E+03
smooth
1E+04
1E+05
1E+06
Re
1E+07
1E+08
Moody Diagram
0.1
 D
f  Cp 
l 

0.05
0.04
0.03
friction factor
0.02
0.015
0.01
0.008
0.006
0.004
e
D
laminar
0.002
0.001
0.0008
0.0004
0.0002
0.0001
0.00005
0.01
1E+03
smooth
1E+04
1E+05
1E+06
Re
1E+07
1E+08
Minor Losses
LSC Pipeline
cs2
z=0
cs1
0
V12 p2
V22
 z1   1
  z2   2
 hl

2g 
2g
p1
Ignore entrance losses
-2.85 m
28 kPa is equivalent to 2.85 m of water
L
F
F
LQ I
L I O
D  0.66M
e G J  Q G J P
Hgh KP
M
N Hgh K
Q
2
1.25
9 .4
f
D  154
. m
5.2 0.04
4 .75
f
Q  2 m3 / s
  106 m2 / s
L  3100m
e  0.002m
h f  2.85m
Directions
 Assume
 find
fully turbulent (rough pipe law)
f from Moody (or from von Karman)
 Find
total head loss (draw control volume)
 Solve for Q using symbols (must include
minor losses) (no iteration required)
hl = å hf + å hminor
Solution
0.1
0.05
0.04
0.03
friction factor
0.02
0.015
0.01
0.008
0.006
0.004
laminar
0.002
0.001
0.0008
0.0004
0.0002
0.0001
0.00005
0.01
1E+03
smooth
1E+04
1E+05
1E+06
Re
1E+07
1E+08
Pipe roughness
Find Q given pipe system
hminor  K
8Q 2
g 2 D 4
8 LQ 2
hf = f
gp 2 D 5
hl = å hf + å hminor
8Q 2 é æ L ö
æK öù
hl =
f 5 +å
2 êå
èD 4 øú
gp ë è D ø
û
Q =p
é
8 êå
ë
ghl
æ L ö
æK öù
f 5 +å
è D ø
èD 4 øú
û