Chapter 4

BFC 10403
FLUID MECHANICS
NOOR ALIZA AHMAD
aliza@uthm.edu.my
Learning Outcome
• At the end of this chapter, students should be able
to:
i. Understanding of laminar and turbulent flow in pipes
and the analysis of fully developed flow
ii. Calculate the major and minor losses associated with
pipe flow in piping networks
iii. Understand the different velocity and flow rate
measurement techniques
Chapter 4: Analysis of Flow in
Pipes
Noor Aliza Ahmad
Department of Water Resources and Environment
Faculty of Civil Engineering and Environmental
UTHM
4.1 Characterization of flow based on Reynolds
number
Osborne Reynolds (a British engineer) conducted a flow
experiment, i.e. by injecting dye into pipe flow to classify
the types of flow.
Near laminar
Turbulent
Laminar and Turbulent Flows
Laminar: Smooth
streamlines and highly
ordered motion.
Turbulent: Velocity
fluctuations and highly
disordered motion.
Transition: The flow
fluctuates between laminar
and turbulent flows.
Most flows encountered in
practice are turbulent.
The behavior of
Laminar and
colored fluid
turbulent flow
injected into the
regimes of
flow in laminar
candle smoke
and turbulent
plume.
flows in a pipe.
BFC10403- FLUID MECHANICS
Noor Aliza Ahmad , aliza@uthm.edu.my
Three types of flow:
Laminar flow
Re < 2000
Transitional flow
2000 < Re < 4000
Turbulent flow
Re > 4000
Reynolds number Re is the ratio of the inertia force on an element of fluid

to the viscous force.
VD VD
Re   Re is dimensionless.
 
where V = average velocity, D = diameter of pipe,  = fluid density, 
= dynamic viscosity, and  = kinematic viscosity
6
BFC 10403

Example 4.1
Determine the range of average velocity of flow for which the flow
would be in the transitional region if an oil of S.G. = 0.89 and dynamic
viscosity = 0.1 Ns/m2 is flowing in a 2-in pipe.
For transitional flow, 2000  Re  4000
VD
Re 


V  Re
D
0.1
When Re = 2000, V  2000 
890  2  0.0254
V  4.424 m/s
0.1
When Re = 4000, V  4000 
890  2  0.0254
V  8.847 m/s
Therefore, for the flow to be in transitional state, the average velocity V should
be between 4.424 m/s and 8.847 m/s.
8
Friction in Pipes of Constant Cross Section
Head loss or energy loss in pipe flow maybe caused by friction, valves, fittings,
expansion and contraction in pipe diameter, and others.
Frictional head loss is known as the major head loss in pipe flow as the loss
occurs along the pipe while other head losses such as due valves, fittings,
expansion and contraction are known as minor losses.
Friction head loss can be estimated using the Darcy-Weisbach equation:

fL V 2
hf 
D 2g
where hf = head loss due to friction, f = friction factor (dimensionless), L =
length of flow (or pipe), D = diameter of pipe, V = average velocity of flow, and g
= gravity acceleration.
For laminar flow, the head loss due to friction can also be estimated
using the Hagen-Poiseuille equation: 32LV
hf 
D 2
where  = dynamic viscosity, L = length of flow (or pipe), D = diameter of
pipe, V = average velocity of flow, and  = specific weight of liquid. 13
For laminar flow, the friction factor f can be calculated based on Reynolds
number.
fL V 2
Darcy-Weisbach equation: hf 
D 2g
32LV
Hagen-Poiseuille equation: hf 
D 2
fLV 2 32LV
Equating both equations: 
2gD D 2
64 
f 
VD
64
f  Friction factor for laminar flow
Re
14
Example 4.2
In a refinery oil (S.G. = 0.85,  = 1.8  105 m2/s) flows through
a 30 m long, 100 mm diameter pipe at 0.50 L/s. Is the flow
laminar or turbulent? Find the head loss per meter of pipe
length. 3
Q 0.5  10
VD V   0.0637 m/s
Reynolds number Re  A    0.12 
  
0.0637  0.1  4 
Re 
1.8  10 5
Re  353 .68 < 2000, therefore laminar flow
64 64
f 
For laminar flow, friction factor   0.1810
Re 353.68
fL V 2 1
Head loss per meter pipe length hf 
D 2g L
0.181 0.06372
hf  
0 .1 2  9.81
hf  3.74  104 m
17
PRESSURE DROP
• The pressure drop ΔP is directly related to the power
requirements of the fan or pump to maintain the flow.
Darcy-Weisbach fL V 2 (laminar and

PL 
equation: D 2 turbulent flow)
Hagen-Poiseuille 32LV (laminar flow)

P 
equation: D2
21
Flowrate (Q) for laminar flow using Hagen-Poiseuille equation:
• Horizontal Pipe:
PD 2 PD 4
Vavg  and Q
32L 128L
 Inclined Pipe:
(P  gL sin )D 2 (P  gL sin )D 4
Vavg  and Q
32L 128L
22
EXAMPLE 4.3(a) (ΔP and H.P)
SAE oil flows 10 L/s through 150 mm diameter and 40 m long
pipe. Assumed the flow is laminar, determine ΔP in the
pipeline using Hagen-Poiseuille Darcy-Weisbach equations
(µ = 8.14 x 10-2 Pa.s, ρ =869 kg/m3)
Q 10x103
V  2
 0.566m / s
A  0.15 
 
 2 
VD  869  x0.566x0.15
Re   2
 906.36  2, 000
 8.14x10
the flow is laminar
23
Hagen-Poiseuille equation:
32LV 32(8.14x102 )(40)(0.566)

P  2
 2
 2.62kPa
D 0.15
Darcy-Weisbach equation:
64 64
Friction factor: f   0.0706
Re 906.36
fL V 2  0.0706  (40) (869)(0.566) 2

PL    2.62kPa
D 2 0.15 2
24
EXAMPLE 4.3(b) (Q and H.P)
Oil is flowing steadily through a 5 cm diameter along 40 m
pipe. The pressure at the inlet and outlet pipe are 745 kPa
and 97 kPa, respectively. Assume the flow is laminar,
determine the flow rate of oil if pipe is
(a) horizontal
(b) inclined 15o upward from horizontal axis
(c) inclined 15o downward from horizontal axis
(ρ = 869 kg/m3 and µ = 0.8 kg/m.s)
25
ΔP = P1 – P2 = 745 – 97 = 648 kPa, D = 5 cm = 0.05 m
(a) horizontal
PD 4 (65k)(0.05) 4
Q   3.11x103 m3 / s
128L 128(0.8)(40)
(b) inclined 15o upward from horizontal axis

(P  gLsin )D 4 (648k  869x9.81x40sin15o )0.054
Q   2.67x103 m3 / s
128L 128(0.8)(40)
(c) inclined 15o downward from horizontal axis
(P  gLsin )D 4 (648k  869x9.81x40sin(15o ))0.054
Q   3.54x103 m3 / s
128L 128(0.8)(40)
26
4.3 Friction factor and Moody Chart
Friction factor can be estimated by the following equations:
64
Darcy-Weisbach & Hagen-Poiseuille: f  for Re < 2000
Re
0.316
Blasius (for smooth pipes): f  for 3000  Re  100000
Re0.25
1  e D 2.51 
Colebrook (for all pipes):  2log   for 4000  Re  108
f  3.7 Re f 
For convenience, these friction factor equations are used to prepare a Moody diagram.
Moody diagram provides the relationship between friction factor f, Reynolds number Re and pipe
relative roughness e/D, where e is the absolute roughness.
Pipe surface roughness

measurement
27
30
4.4

4.4

4.4

f = 0.021
ε= 0.001
Re = 3.72 x 105
Example 4.5
Determine the friction factor f if water at 70C is flowing at 9.14
m/s in stainless steel pipe having an inside diameter of 25 mm.
Given velocity of flow = 9.14 m/s and diameter of pipe D = 0.025 m
For water at 70C, kinematic viscosity  = 9.75  107 m2/s
Reynolds number Re  VD  9.14  0.025  2.344  105 > 4000,

 9.75  10 7 turbulent
Stainless steel pipe, e = 0.002 mm flow
e 0.002
Relative roughness   0.00008
D 25
From Moody diagram, f  0.0159
36
f = 0.0159
0.00008
2.34  105
37
BFC 10403

Minor Losses
• Piping systems include fittings, valves, bends, elbows, tees,
inlets, exits, enlargements, and contractions.
• These components interrupt the smooth flow of fluid and
cause additional losses because of flow separation and mixing
• We introduce a relation for the minor losses associated with
these components
• KL is the loss coefficient.

• Is different for each component.
• Is assumed to be independent of Re.
• Typically provided by manufacturer or
generic table (e.g., Table 8-4 in text).
a. Loss of head at entrance
V2
Loss due to the entrance of flow into pipe he  k e
2g
where V = mean velocity in the pipe, ke = loss coefficient
ke  0.5
56
b. Loss of head at submerged discharge
Loss of head when pipe discharges fluid into filled V2

hd  kd
reservoir or tank 2g
kd  1.0
57
c. Loss of head due to contraction
V22
Loss of head when flow contracts hc  k c
2g Loss coefficients for sudden contraction
D2
kc
D1
0.0 0.50
0.1 0.45
0.2 0.42
0.3 0.39
0.4 0.36
0.5 0.33
0.6 0.28
0.7 0.22
0.8 0.15
0.9 0.06
1.0 0.00
58
V22
Loss of head when flow contracts hc  k c
2g
For smooth curved transition, kc can be as small as

0.05.
For conical transducer with angle 20 to 40, a minimum kc is
about 0.10.
59
d. Loss of head due to expansion
Loss of head when flow expands.
hx 
V1  V2 
2
Loss of head due to sudden expansion

2g
Loss of head due to gradual expansion hx  k x

V1  V2 2
2g
60
Loss of head due to gradual expansion hx  k x
V1  V2 2
2g
kx
kx
61
e. Loss of head due to pipe fittings
Loss of head due to pipe fittings such as valve, bend, and V2
elbow hk
2g
Values of loss factors for pipe fittings
Fitting k
Globe valve, wide open 10
Angle valve, wide open 5
Close-return bend 2.2
T, through side outlet 1.8
Short-radius elbow 0.9
Medium-radius elbow 0.75
Long-radius elbow 0.60
45 elbow 0.42
Gate valve, wide open 0.19
Gate valve, half open 2.06
62
f. Loss of head in bends and elbows
V2
Loss of head due to bends and elbows hb  k b
2g
Loss coefficient due to 90 bend
63
V2
hb  k b
2g

Example 4.12
Water at 15C is being pumped from a stream to a reservoir whose surface is 64 m above
the pump. The pipe from the pump to the reservoir is 8 in steel pipe 762 m long. If 0.113
m3/s is being pumped, compute the pressure at the outlet of the pump. Consider the
friction loss in the discharge line, but neglect other losses.
If the pressure at the pump inlet is 16.27 kPa, compute the power delivered by the pump
to the water.
Water at 15C,  =1000 kg/m3,  = 1.15  106 m2/s B pB = 0, VB = 0

zB  zA = 64 m

D = 8 in = 0.2032 m
L = 762 m
Steel pipe, e = 0.045 mm
e 0.045
  0.000221
D 203.2
 A
Q = 0.113 m3/s
V  3.485 m/s
70
B
VD
Re 

3.485  0.2032

1.15  10 6
 6.1578  105
A
From Moody chart, f = 0.0158
pA VA2 pB VB2
Energy equation between A and B:  zA   hf   zB 
g 2g g 2g
pA VA2 fL V 2
 zA    zB
g 2g D 2g
pA 3.485 2 0.0158  762 3.485 2

    64
9810 2  9.81 0.2032 2  9.81
pA  981.6 kPa
71
Power delivered by pump to water
Ppump  hpump Q
poutlet  981.6 kPa
Ppump  ppump Q
pinlet  16.27 kPa

Ppump  981.6   16.27   103  0.113
Ppump  112.8 kW
72
THANK YOU

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BFC 10403

Reynolds number Re is the ratio of the inertia force on an element of fluid

BFC10403- FLUID MECHANICS

Friction head loss can be estimated using the Darcy-Weisbach equation:

Darcy-Weisbach fL V 2 (laminar and

Hagen-Poiseuille 32LV (laminar flow)

32LV 32(8.14x102 )(40)(0.566)

fL V 2  0.0706  (40) (869)(0.566) 2

(b) inclined 15o upward from horizontal axis

Pipe surface roughness

BFC10403- FLUID MECHANICS

BFC10403- FLUID MECHANICS

BFC10403- FLUID MECHANICS

Reynolds number Re  VD  9.14  0.025  2.344  105 > 4000,

From Moody diagram, f  0.0159

BFC10403- FLUID MECHANICS

• KL is the loss coefficient.

Loss of head when pipe discharges fluid into filled V2

For smooth curved transition, kc can be as small as

Loss of head when flow expands.

Loss of head due to sudden expansion

Loss of head due to gradual expansion hx  k x

Loss coefficient due to 90 bend

BFC10403- FLUID MECHANICS

Water at 15C,  =1000 kg/m3,  = 1.15  106 m2/s B pB = 0, VB = 0

pA 3.485 2 0.0158  762 3.485 2

pinlet  16.27 kPa

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