Fluid Mechanics

1. Dynamics of Viscous Fluid Flow in
Closed Pipe: Darcy-Weisbach equation for
flow in pipes. Major and minor losses in pipe
lines.
Dr. Mohsin Siddique
Assistant Professor
1
FLUID MECHANICS

2. Steady Flow Through Pipes
2
Laminar Flow:
flow in layers
Re<2000 (pipe flow)
Turbulent Flow:
flow layers mixing with each
other
Re >4000 (pipe flow)

3. Steady Flow Through Pipes
3
Reynold’s Number(R or Re): It is ratio of inertial forces (Fi) to
viscous forces (Fv) of flowing fluid
For laminar flow: Re<=2000
For transitional flow: 2000<Re<4000
For Turbulent flow: Re>= 4000
. .
Re
. .
. . .
. . .
Velocity Volume
Mass Velocity
Fi Time Time
Fv Shear Stress Area Shear Stress Area
Q V AV V AV V VL VL
du VA A A
dy L
ρ
ρ ρ ρ ρ
τ µ υµ µ
= = =
= = = = =
Where ;
V is avg. velocity of flow in pipe
ν is kinematic viscosity
L is characteristic/representative
linear dimension of pipe. It is
diameter of pipe (circular conduits)
or hydraulic radius (non-circular
conduits).
νµ
ρ VDVD
Re ==
Note: For non-circular section, we need to use hydraulic radius (Rh) instead
of diameter (D) for the linear dimension (L).
Values of critical
Reynolds no.

4. Steady Flow Through Pipes
4
Hydraulic Radius (Rh) or Hydraulic
Diameter: It is the ratio of area of flow
to wetted perimeter of a channel or pipe
P
A
perimeterwetted
Area
Rh ==
For Circular Pipe
( )( )
h
h
RD
D
D
D
P
A
R
4
4
4/ 2
=
===
π
π
For Rectangular pipe
DB
BD
P
A
Rh
2+
==
B
D
Note: hydraulic Radius gives us indication for most economical section. More
the Rh more economical will be the section.
νν
h
h
VRVD
R
4
== By replacing D with Rh, Reynolds’ number formulae
can be used for non-circular sections as well.

5. Head Loss in Pipes
5
Total Head Loss=Major Losses+ Minor Losses
Major Loss: Due to pipe friction
Minor Loss: Due to pipe fittings, bents and valves etc

6. Head Loss in Pipes due to Friction
6
The head loss due to friction in a
given length of pipe is proportional
to mean velocity of flow (V) as long
as the flow in laminar. i.e.,
But with increasing velocity, as the
flow become turbulent the head
loss also varies and become
proportion toVn
Where n ranges from 1.75 to 2
Log-log plot for flow in uniform pipe
(n=2.0 for rough wall pipe; n=1.75 for
smooth wall pipe
VH f ∝
n
f VH ∝

7. Frictional Head Loss in Conduits of Constant
Cross-Section
7
Consider stead flow in a conduit of uniform cross-section A.The pressure
at section 1 & 2 are P1 & P2 respectively.The distance between the section
is L. For equilibrium in stead flow,
∑ == 0maF
Figure: Schematic diagram of conduit
0sin 21 =−−− APPLWAP oτα
P= perimeter of conduit
= Avg. shear stress
between pipe boundary
and liquid
oτ
012
21 =−




 −
−− PL
L
zz
ALAPAP oτγ
αsin12
=




 −
L
zz

8. Frictional Head Loss in Conduits of Constant
cross-section
8
012
21 =−




 −
−− PL
L
zz
ALAPAP oτγ
Dividing the equation by Aγ
( ) 012
21
=−−−−
A
PL
zz
PP o
γ
τ
γγ
fL
o
hh
A
PLP
z
P
z ===





+−





+
γ
τ
γγ
2
2
1
1
Therefore, head loss due to friction hf can
be written as
h
oo
f
R
L
A
PL
h
γ
τ
γ
τ
==
Lh
g
v
z
P
g
v
z
P
+++=++
22
2
2
2
2
2
1
1
1
γγ
Remember !! For pipe flow
For stead flow in pipe of
uniform diameter v1=v2
Lhz
P
z
P
=





+−





+ 2
2
1
1
γγ
This is general equation and can be applied to any shape conduit having
either Laminar or turbulent flow.
P
A
Rh =Q

9. Determining Shear Stress
9
For smooth-walled pipes/conduits, the average shear stress at the
wall is
Using Rayleigh's Theorem of dimensional analysis, the above relation
can be written as;
Rewriting above equation in terms of dimension (FLT), we get
( ),,,, VRf ho ρµτ =
( ) 

























=
ncb
a
T
L
L
FT
L
FT
LK
L
F
4
2
22
( )ncba
ho VRk ... ρµτ =
LlengthR
L
F
areaforce
h
o
==
== 2
/τ
( )
22
4233
/.
////
/
−
==
===
=
FTLmsN
LFTLaFLM
TLV
µ
ρ( ) ( ) ( ) ( )( )ncba
TLLFTFTLLKFL /4222 −−−
=

10. Determining Shear Stress
10
According to dimensional homogeneity, the dimension must be equal on
either side of the equation, i.e.,
( ) ( ) ( ) ( )( )ncba
TLLFTFTLLKFL /4222 −−−
=
)(20:
)(422:
)(1:
iiincbT
iincbaL
icbF
→−+=
→+−−=−
→+= Solving three equations, we get
1;2;2 −=−=−= ncnbna
Substituting values back in above equation
( ) ( ) 2
2
122
...... V
VR
kVRkVRk
n
hnnnn
h
ncba
ho ρ
µ
ρ
ρµρµτ
−
−−−






===
( ) 22
VRk
n
eo ρτ −
=
Setting = we get
2
2
V
Cfo ρτ = Where, Cf is coefficient of friction
( ) 2−n
eRk 2/fC

11. Determining Shear Stress
11
Now substituting the equation of avg. shear stress in equation of head loss,
For circular pipe flows, Rh=D/4
Where, f is a friction factor. i.e.,
The above equation is known as pipe friction equation and as Darcy-
Weisbach equation.
It is used for calculation of pipe-friction head loss for circular pipes
flowing full (laminar or turbulent)
2/2
VCfo ρτ =
h
o
f
R
L
h
γ
τ
=
h
f
h
f
f
gR
LVC
R
LVC
h
22
22
==
γ
ρ
g
V
D
L
f
g
V
D
L
C
Dg
LVC
h f
f
f
22
4
42
4 222
===
( )Re4 fCf f ==

12. Friction Factor for Laminar and
Turbulent Flows in Circular Pipes
12
Smooth and Rough Pipe
Mathematically;
Smooth Pipe
Rough Pipe
Transitional mode
Turbulent flow near boundary
=
=
v
e
δ
Roughness height
Thickness of viscous sub-layer
Smooth pipe
Rough pipe
f
D
fV
v
Re
14.1414.14
==
ν
δ
ve δ<
ve δ>
ve δ<
ve δ14>
vv e δδ 14≤≤

13. Friction Factor for Laminar and Turbulent Flows in
Circular Pipes
13
For laminar flow
For turbulent flow
Re
64
=f2000Re <
51.2
Re
log2
1 f
f
=
9.6
Re
log8.1
1
=
f
25.0
Re
316.0
=f
7/1
max






=
or
y
u
u
Def /
7.3
log2
1
=








+−=
f
De
f Re
51.2
7.3
/
log2
1








+





−=
Re
9.6
7.3
/
log8.1
1
11.1
De
f
Colebrook Eq. for turbulent flow in all pipes
Halaand Eq. For turbulent flow in all pipes
Von-karman Eq. for fully rough flow
Blacius Eq. for smooth pipe flow
Seventh-root law
5
10Re3000 ≤≤
From Nikuradse experiments
Colebrook Eq. for smooth pipe flow
for smooth
pipe flow
4000Re >

14. Friction Factor for Laminar and Turbulent Flows
in Circular Pipes
14
The Moody chart or Moody diagram is a graph in non-
dimensional form that relates the Darcy-Weisbach friction factor, Reynolds
number and relative roughness for fully developed flow in a circular pipe.
The Moody chart is universally valid for all steady, fully developed,
incompressible pipe flows.

15. Friction Factor for Laminar and Turbulent Flows
in Circular Pipes
15
For laminar flow For non-laminar flow
eR
f
64
= 







+−=
f
De
f Re
51.2
7.3
/
log2
1
Colebrook eq.

16. Friction Factor for Laminar and
Turbulent Flows in Circular Pipes
16
The friction factor can be determined by its Reynolds number (Re) and the
Relative roughness (e/D) of the Pipe.( where: e = absolute roughness and D
= diameter of pipe)

17. 17

18. Problem Types
18
Type 1: Determine f and hf,
Type 2: Determine Q
Type 3: Determine D

19. Problem
19
Find friction factor for the following pipe
e=0.002 ft
D=1ft
KinematicViscosity, ν=14.1x10-6ft2/s
Velocity of flow,V=0.141ft/s
Solution:
e/D=0.002/1=0.002
R=VD/ ν =1x0.141/(14.1x10-6)=10000
From Moody’s Diagram; f=0.034
___________
Re
51.2
7.3
/
log2
1
=








+−=
f
f
De
f

20. Problem-Type 1
20
Pipe dia= 3 inch & L=100m
Re=50,000 ʋ=1.059x10-5ft2/s
(a): Laminar flow:
f=64/Re=64/50,000=0.00128
ft
gD
fLV
HLf 0357.0
)12/3)(2.32(2
)12.2)(100(00128.0
2
22
===
sftV
VVD
/12.2
10059.1
)12/3(
50000Re 5
=⇒
×
=⇒= −
ν

21. Problem-Type 1
21
Pipe dia= 3 inch & L=100m
Re=50,000 ʋ=1.059x10-5ft2/s
(b): Turbulent flow in smooth pipe: i.e.: e=0
0209.0
50000
51.2
7.3
0
log2
Re
51.2
7.3
/
log2
1
=








+−=








+−=
f
ff
De
f
ft
gD
fLV
HLf 582.0
)12/3)(2.32(2
)12.2)(100(0209.0
2
22
===

22. Problem-Type 1
22
Pipe dia= 3 inch & L=100m
Re=50,000 ʋ=1.059x10-5ft2/s
(c): Turbulent flow in rough pipe: i.e.: e/D=0.05
0720.0
50000
51.2
7.3
05.0
log2
Re
51.2
7.3
/
log2
1
=








+−=








+−=
f
ff
De
f
ft
gD
fLV
HLf 01.2
)12/3)(2.32(2
)12.2)(100(0720.0
2
22
===

23. Problem-Type 1
23
hL=?
memDmL 0005.0;25.0;1000 ===
smsmQ /10306.1;/051.0 263 −
×== ν
( ) ( )
002.025.0/0005.0/
10210306.1/25.0039.1/ 56
==
×=××== −
De
VDR ν
0.0245f
DiagramsMoody'From
=
m
gD
fLV
hL 39.5
2
2
==
smAQV /039.1/ ==Q

24. Problem-Type 2
24
gDfLVhL 2/2
=

25. Problem-Type 2
25
For laminar flow For non-laminar flow
eR
f
64
= 







+−=
f
De
f Re
51.2
7.3
/
log2
1
Colebrook eq.

26. Problem-Type 3
26

27. Problem
27
gD
flV
h
f
De
f
Lf
2
Re
51.2
7.3
/
log2
1
2
=








+−=

28. Problem
28

29. Problem
29

30. MINOR LOSSES
30
Each type of loss can be quantified using a loss coefficient (K).
Losses are proportional to velocity of flow and geometry of device.
Where, Hm is minor loss and K is minor loss coefficient.The value of
K is typically provided for various types/devices
NOTE: If L > 1000D minor losses become significantly less
than that of major losses and hence can be neglected.
g
V
KHm
2
2
=

31. Minor Losses
31
These can be categorized as
1. Head loss due to contraction in pipe
1.1 Sudden Contraction
1.2 Gradual Contraction
2. Entrance loss
3. Head loss due to enlargement of pipe
3.1 Sudden Enlargement
3.2 Gradual Enlargement
4. Exit loss
5. Head loss due to pipe fittings
6. Head loss due to bends and elbows

32. Minor Losses
32
Head loss due to contraction of pipe (Sudden contraction)
A sudden contraction (Figure) in pipe usually causes a marked drop
in pressure in the pipe because of both the increase in velocity and
the loss of energy of turbulence.
g
V
KH cm
2
2
2
=
Head loss due to sudden contraction is
Where, kc is sudden contraction
coefficient and it value depends
up ratio of D2/D1 and velocity
(V2) in smaller pipe

33. Minor Losses
33
Head loss due to enlargement of pipe (Gradual Contraction)
Head loss from pipe contraction may be greatly reduced by
introducing a gradual pipe transition known as a confusor as shown
Figure.
g
V
KH cm
2
'
2
2
=
Head loss due to gradual
contraction is
Where, kc
’ is gradual contraction
coefficient and it value depends
up ratio of D2/D1 and velocity
(V2) in smaller pipe

34. Minor Losses
34
Entrance loss
The general equation for an entrance head loss is also expressed in
terms of velocity head of the pipe:
The approximate values for the entrance loss coefficient (Ke)
for different entrance conditions are given below
g
V
KH em
2
2
=

35. Minor Losses
35
head loss due to enlargement of pipe (Sudden Enlargement)
The behavior of the energy grade line and the hydraulic grade line in
the vicinity of a sudden pipe expansion is shown in Figure
The magnitude of the head
loss may be expressed as
( )
g
VV
Hm
2
2
21 −
=

36. Minor Losses
36
head loss due to enlargement of pipe (Gradual Enlargement)
The head loss resulting from pipe expansions may be greatly
reduced by introducing a gradual pipe transition known as a diffusor
The magnitude of the head
loss may be expressed as
( )
g
VV
KH em
2
2
21 −
=
The values of Ke’ vary with the diffuser angle (α).

37. Minor Losses
37
Exit Loss
A submerged pipe discharging into a large reservoir (Figure ) is a
special case of head loss from expansion.
( )
g
V
KH dm
2
2
=
Exit (discharge) head loss is
expressed as
where the exit (discharge) loss
coefficient Kd=1.0.

38. Minor Losses
38
Head loss due to fittings valves
Fittings are installed in pipelines to control flow.As with other losses in
pipes, the head loss through fittings may also be expressed in terms of
velocity head in the pipe:
g
V
KH fm
2
2
=

39. Minor Losses
39
Head loss due to bends
The head loss produced at a bend was found to be dependent of the
ratio the radius of curvature of the bend (R) to the diameter of the
pipe (D).The loss of head due to a bend may be expressed in terms
of the velocity head as
For smooth pipe bend of 900, the values of Kb for various values of
R/D are listed in following table.
g
V
KH bm
2
2
=

40. Minor Losses
40

41. Numerical Problems
41

42. Numerical Problems
42

43. Thank you
Questions….
Feel free to contact:
43