chapter 4 fluid mechanics
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fluids mechanicsTRANSCRIPT
FLUID MECHANICSCLD 10603
INTERNAL FLOW IN PIPES AND CONDUITS
CLD 10603 Chapter 4: Internal Flow in Pipe & Conduits
Steam pipe bridge in a geothermal power plant. Pipe flows are everywhere, often occurring in groups or networks. They are designed using the principles outlined in this chapter.
CLD 10603 3 Chapter 4: Internal Flow in Pipe & Conduits
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Student should be able to: Identify the characteristics of laminar flow and
turbulent flow Employ Poiseulle and Darcy equation in basic
problems Recall Bernoulli’s equation Employ Bernoulli’s equation in flow measurement Identify major losses Identify minor losses
Objectives
The three regimes of viscous flow:
a) Laminar flow at low Re;
b) Transition at intermediate Re;
c) Turbulent flow at high Re
CLD 10603 4 Chapter 4: Internal Flow in Pipe & Conduits
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Types of Flow
Laminar Flow Flow structure is characterized by smooth
motion in lamina or layer. There is no macroscopic mixing of adjacent fluid layer.
CLD 10603 5 Chapter 4: Internal Flow in Pipe & Conduits
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Types of Flow
- E.g. : A thin filament of dye injected into a laminar flow appears as a single line. There is no dispersion of dye throughout the flow, except the slow dispersion due to molecular motion.
Laminar Flow
- Re < 2000
- 'low' velocity
- Dye does not mix with water
- Fluid particles move in straight lines
- Simple mathematical analysis possible
- Rare in practice in water systems.
CLD 10603 6 Chapter 4: Internal Flow in Pipe & Conduits
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Types of Flow
Turbulent Flow Flow structure is characterized by random,
three-dimensional motion of fluid particles superimposed on the mean motion.
CLD 10603 7 Chapter 4: Internal Flow in Pipe & Conduits
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Types of Flow
- E.g.: If a dye filament injected into a turbulent flow, it disperses quickly throughout the flow field; the line of dye breaks up into myriad entangled threads of dye.
Turbulent Flow
- Re > 5000
- 'high' velocity
- Dye mixes rapidly and completely
- Particle paths completely irregular
- Average motion is in the direction of the flow
CLD 10603 8 Chapter 4: Internal Flow in Pipe & Conduits
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Types of Flow
Turbulent Flow - Cannot be seen by the naked eye - Changes/fluctuations are very difficult to
detect. Must use laser. - Mathematical analysis very difficult - so
experimental measures are used - Most common type of flow.
CLD 10603 9 Chapter 4: Internal Flow in Pipe & Conduits
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Types of Flow
Laminar Turbulent
Re < 2000 Re > 4000
Inefficient mixing – low radial mixing
Radial mixing can be enhanced
Broad residence time distribution
Short residence time distribution
Low power High power
CLD 10603 10 Chapter 4: Internal Flow in Pipe & Conduits
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Types of Flow
The primary parameter correlating the viscous behaviour of all Newtonian fluids.
The ratio of inertial forces to viscous forces.
Where V = velocity (m/s), D = diameter (m),
is kinematic viscosity (m2/s).
CLD 10603 10 Chapter 4: Internal Flow in Pipe & Conduits
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Reynolds Number
uDVD
Re
Used to determine whether a flow is laminar or turbulent.
Reynolds Number is dimensionless. Re < 2000: Laminar Flow Re between 2000 to 4000 : Transitional
Flow Re > 4000 : Turbulent Flow
CLD 10603 12 Chapter 4: Internal Flow in Pipe & Conduits
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Reynolds Number
Question 1
Water flow through a pipe 25 mm in diameter at a velocity of 6 m/s. Determine whether the flow will be laminar or turbulent assuming that the viscosity of water is 1.30 x 10-3 kgm-1s-1 and its density 1000 kgm-3.
CLD 10603 13 Chapter 4: Internal Flow in Pipe & Conduits
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Reynolds Number
Question 2
If oil of S.G 0.9 and viscosity 9.6 x 10-2 kgm-1s-1 is pumped through the same pipe, what type of flow will occur?
CLD 10603 15 Chapter 4: Internal Flow in Pipe & Conduits
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Reynolds Number
The total quantity of fluid flowing in unit time past any particular cross section of a stream is called the discharge or flow at that section.
It can be measured either in terms of:
i. volume (volumetric flowrate = Q = Au)
Unit : m3/s
ii. mass (mass flowrate = = Qρ).
Unit : kg/s
CLD 10603 17 Chapter 4: Internal Flow in Pipe & Conduits
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Discharge
m
Question 3
Water flowing at the rate of 10 gal/min in pipe having an inside diameter of 0.0525 m. Calculate the Reynolds Number if = 8.007 x 10-4 Pa.s. Given H2O = 996 kg/m3.
CLD 10603 18 Chapter 4: Internal Flow in Pipe & Conduits
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Reynolds Number
The Bernoulli’s equation states that:
P + ½ u2 + gz = constant
Where:
P = pressure
= density
g = acceleration of gravity
z = height
u = velocity
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 34
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Bernoulli's Equation
It may be written as:
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 35
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Bernoulli's Equation
2
222
1
211
22z
g
u
g
pz
g
u
g
p
Bernoulli's equation has some restrictions in its applicability, they are: Flow is steady; Density is constant (incompressible fluid) Friction losses are negligible. The equation relates the states at two
points along a single streamline
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 36
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Bernoulli's Equation
Flow through a small orifice
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 40
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Bernoulli's Application
Orifice
A actual
Vena contractor
u1 = 0 (negligible)
P1 = 0 (P atm)
P2 = 0 (P atm)
Take z2 as the datum z1= h, z2= 0
2
222
1
211
22z
g
u
g
pz
g
u
g
p
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 41
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Bernoulli's Application
g
uz
2
22
1
2
222
1
211
22z
g
u
g
pz
g
u
g
p
12 2gzu
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 42
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Bernoulli's Application
Discharge through the orifice is given by:
Q = Au
Q actual = A actual u actual
= cd A orifice u theoretical
= cd A orifice 2gh
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 43
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Flow Measurement - Orifice
Question 5
A 100 mm diameter standard orifice discharges water under a 6.1 m head. What is the volumetric flow rate? Given cd = 0.594
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 44
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Flow Measurement - Orifice
Question 6
The actual velocity in the contracted section of a jet of liquid flowing from a 50 mm diameter orifice is 8.53 ms-1 under a head of 4.57 m. If the measured discharge is 0.0114 m3s-1,what is the value of the coefficient of discharge (cd)?
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 46
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Flow Measurement - Orifice
A device to measure the local velocity along a streamline.
u2 = 0 (stagnation point)
z1=z2= 0
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 37
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Flow Measurement – Pitot Tube
2
222
1
211
22z
g
u
g
pz
g
u
g
p
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 38
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Flow Measurement – Pitot Tube
2
222
1
211
22z
g
u
g
Pz
g
u
g
P
)(2
)(2
2
121
1221
2112
PPu
PPu
g
u
g
PP
Where:P = h(man -)g
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 39
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Flow Measurement – Pitot Tube
)(21 P
u
A device for measuring discharge in a pipe.
Consists of a rapidly converging section which increases the velocity of flow and hence reduces the pressure.
Then returns to the original dimensions of the pipe by a gently diverging 'diffuser' section.
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 48
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Flow Measurement - Venturi
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 49
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Flow Measurement - Venturi
A Venturi Meter
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 50
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Flow Measurement - Venturi
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 50
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Flow Measurement - Venturi
Discharge can be expressed in terms of the manometer reading , giving:
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 52
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Flow Measurement - Venturi
22
21
21
12
AA
gh
AACQ
man
dactual
Question 7
A Venturi meter is being used to measure flow in a pipeline of diameter 250 mm which carries water. When the pressure difference between the throat and the entrance of the Venturi meter is 300 mm on a mercury manometer, determine the flow in the pipeline. The Venturi meter has a throat diameter of 80 mm and a coefficient of discharge of 0.97. The relative density of mercury is 13.6.
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 53
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Flow Measurement - Venturi
Question 8
A Venturi meter measures the flow speed of water through a pipe. The speed to be measured is v1, the speed in the pipe where the cross-sectional area is A1 = 4.10 × 10–4 m2. Pressure meters measure the pressure in this pipe (P1), and the pressure (P2) at a narrow section of pipe with cross-sectional area A2 = 3.24 × 10–4 m2. Assume water behaves as an ideal, non-viscous fluid. (Density of water is 1000 kg/m3.) If the pressure difference, P1 – P2 is measured to be 81.0 Pa, calculate the speed v1 of the water in the pipe.
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 55
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Flow Measurement - Venturi
CLD 10603 3 Chapter 4: Internal Flow in Pipe & Conduits
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Roughness Friction factor Darcy Weisbach Equation Hagen Poiseuille Law Energy Equation Major head loss Minor head loss
Internal Flow
Relative pipe roughness is the ratio of the pipe surface roughness () to its diameter (D).
Relative pipe roughness = /D
CLD 10603 20 Chapter 4: Internal Flow in Pipe & Conduits
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Roughness
CLD 10603 21 Chapter 4: Internal Flow in Pipe & Conduits
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Roughness
Types of Roughness
The loss of pressure or head due to the resistance to flow in the pipe and fittings. Friction loss is influenced by pipe size and fluid velocity, and is usually expressed in feet of head.
CLD 10603 22 Chapter 4: Internal Flow in Pipe & Conduits
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Friction Factor
The friction coefficient (f), depends on the flow (laminar, transient or turbulent ) and the roughness of the tube or duct.
For fully developed laminar flow the roughness of the duct or pipe can be neglected.
f = 64/Re
CLD 10603 23 Chapter 4: Internal Flow in Pipe & Conduits
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Friction Factor
For hydraulically smooth pipes (= 0) in turbulent flow:
f = 0.3164 / Re 0.25
For rough pipe in turbulent flow, use the Moody Diagram to obtain f.
CLD 10603 24 Chapter 4: Internal Flow in Pipe & Conduits
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Friction Factor
CLD 10603 25 Chapter 4: Internal Flow in Pipe & Conduits
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Friction Factor
Proposed by Julius Weisbach, a German professor who in 1850 published the first modern textbook on hydrodynamics.
An important and widely used equation in hydraulics.
It enables calculation of the head loss due to friction within a given run of pipe.
Valid for duct flows of any cross section and for laminar and turbulent flow.
CLD 10603 25 Chapter 4: Internal Flow in Pipe & Conduits
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CLD 10603 26
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Darcy Weisbach Equation
Where:
hl = head loss
f = friction factor
L = pipe length
D = pipe diameter
V = average flow velocity
g = acceleration of gravity
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 27
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Darcy Weisbach Equation
g
V
D
Lfhl 2
2
2
2
2gA
Q
D
Lf
Question 4
Oil of absolute viscosity 0.101 N.s/m2 and S.G 0.850 flows through 3000 m of 300 mm cast iron pipe at the rate of 0.0444 m3/s.
1) What is the head loss in the pipe?
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 32
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hf = Friction @ Major head loss
hm = Minor head loss
mf hhzg
u
g
pz
g
u
g
p 2
222
1
211
22
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 41
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Energy Equation
- Head loss due to the friction of fluid flowing through a length of pipe.
- For Laminar Flow
- For Turbulent Flow
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 57
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Major Losses
g
V
D
Lhl 2
642
g
V
D
Lfhl 2
2
Energy losses that come from fittings, entrances and exits.
E.g.:
i. Valves and pipe bends
ii. Pipe entrances (for fluid flowing from a tank to a pipe) and pipe exits (fluid flowing from a pipe to a tank).
Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 58
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Minor Losses
g
VKh LL 2
2