viscous flow in pipes
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سمهيثTRANSCRIPT
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FLUID MECHANICS
VISCOUS FLOW IN PIPE
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Learning Outcomes
Characterize flows in pipes.
Explain laminar and turbulent flows and examine their differences
calculate losses in various segments of pipes
Upon the completion of this lecture, you will beable to:
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calculate losses in various segments of pipes
apply appropriate equations and principles to analyze a variety of
pipe flow situations.
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4.1 Flow in Pipe
Pipe is a closed conduit through which fluid may betransported
Behaviour of fluid in pipe has wide variety of reallife applications ranging from large man-made pipesuch as the 800-miles Alaskan pipeline that carries
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crude oil in such as a long distance to the naturalpipes that transports blood throughout humanbodies into and out of their lungs.
In fact numerous applications exist such as housingpipeline network, machinery pipes etc.
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4.1 Flow in Pipe
Knowing there are this variety of applications, it isextremely important to understand thecharacteristic behaviour of moving fluid throughpipe.
Pipe fluid carrierFittings connectors for a desired configuration
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Fittings connectors for a desired configuration of pipes
Pump as energy adderValve flowrate controller
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4.2 General Characteristics of Pipe Flow
Conduits are not necessarily round cross-section but most ofthe common one such as water pipes, hydraulic hoses,and other conduits that are designed to withstand aconsiderable pressure difference across their wallswithout undue distortion of their shape.
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Typical conduits of noncircular cross section includeheating and air conditioning ducts that are often ofrectangular cross section.
Normally the pressure difference between the inside andoutside of these ducts is relatively small. Hence, most of thebasic principles involved are independent of the cross-sectional shape, unless otherwise specified.
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4.2 General Characteristics of Pipe Flow
We consider a pipe completed filled with fluid as shownbelow (a).
We will look at the types of flow such as laminar, transitionalturbulent.
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(a) Pipe flow (b) Open channel flow
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Osborne Reynolds (1842 1912), a British scientist and mathematicianclassified these flows by using a simple apparatus below
He injected dye into a pipe in which water flowed due to gravity. Theentrance region of the pipe is depicted below.
Neutrally buoyant dye is injected, as shown, into the flowing fluid of
4.2 General Characteristics of Pipe Flow
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Neutrally buoyant dye is injected, as shown, into the flowing fluid ofgiven velocity V and pipe diameter D and a streaklike dye shape isformed (only in small enough flowrate)
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VDRe
4.2 General Characteristics of Pipe Flow
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VDRe
4.2 General Characteristics of Pipe Flow
Reynolds Number Flow
Re < 2300 Laminar
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2300 < Re O 4000 Transitional
Re > 4000 Turbulent
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Example 4.1
a. Evaluate the minimum time required to fillthe 1-liter bottle as shown if the flow is (i)Laminar (ii) Turbulent
b. Repeat a for 140oF temperature of thewater.
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4.3 Fully Developed Flow
Fully developed flow
Each fluid particle moves at a constant axial velocity along a streamline The velocity profile u(r) remains unchanged in the flow direction. There is no motion in the radial direction. The velocity component in the direction normal to flow is everywhere zero. There is no acceleration since the flow is steady and fully developed.
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4.3 Pressure Distribution
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4.4 Laminar Flow
Laminar flow exhibits parallel streamlines
Between parallel surfaces, it can be considered that laminar flow is made
up of parallel layers that do not mix up at low average velocity
The Reynolds number indicates that a flow can be laminar, translational or
turbulent as a function of velocity, pipe diameter and viscosity
VDRe
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There must be the critical velocity, diameter and viscosity
VDRe
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4.4 Laminar Flow
Fully developed Laminar Flow
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4.4 Laminar Flow
Fully developed Laminar Flow
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Shear stress distribution within the fluid in a pipe (laminar or turbulentflow) and typical velocity profiles
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4.4 Laminar Flow
rL
P
2
Shear Stress as a function of r
R
r
L
pRrv
2
2
14
)(
Velocity as a function of r
1
2
Average Velocity
L
pD
L
pRv
22
ave328
4
Volumetric Flow rate or Flow rate
pRpD
Q44
5
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RL 4
L
pRv
2
max4
Maximum Velocity at r = 0
3
LL
Q8128
5
Head Loss
2g
v
D
Lfh
gD
Lvh
2
2
L
L
326
7
Hagen Poiseuille Equation
Darcy Energy Loss Equation
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Example 4.2
A 200-m-long pipe made of 3-cm-diamter copper is used to
transport water at 5 liters / second (L/s) and at 4oC having viscosity
of 1.5028x10-3 kg/m.s. Determine the
a) average velocity of the water
b) friction factor
c) pressure drop
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d) head loss
e) required pump power to overcome the head loss.
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Example 4.3
Gasoline of density 680 kg/m3 and viscosity of 3.1x10-4 N.s/m2 flows
is to be transported in a smooth pipe of 40-mm diameter at a rate of
0.001m3/s. determine the ratio of turbulence and laminar head
losses to avoid the turbulence to occur.
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4.5 Laminar vs Turbulent Flows
The fundamental difference between the laminar and turbulent flows is
that the laminar flow does not depend on the pipe wall surface
roughness thus the friction factor is constant (64/Re) regardless of the
relative roughness /D.
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The turbulent flow is dependent on both density and the surface pipe
wall surface roughness. Thus the equation 7
2g
v
D
Lfh
2
L Re64 )D/(Re,f
(laminar) (Turbulent)
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4.6 Minor and Major Losses
When fluid flows through pipe resistance against it exists in
various forms, thus there are pressure drops along the length of
the pipe.
This pressure drop is termed as loss. The loss is divided into two categories
namely minor and major losses
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Minor Losses these losses occur in the following smooth flow interruptions
a. Inlets or exits
b. Sudden enlargement and contraction in a pipe.
c. Bends in a pipe.
d. Any other source of restriction such as pipe fittings and valves.
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4.6 Minor and Major Losses
When fluid flows through pipe resistance against it exists in
various forms, thus there are pressure drops along the length of
the pipe.
This pressure drop is termed as loss. The loss is divided into two categories
namely minor and major losses
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Major Losses these losses are due to the shearing resistance on the pipe
wall surface. Therefore, equation 7 defines the Major Loss or simply
is the Major Head Loss
and the total head loss in the pipe should be the
Minor Head Losses + Major Lead Losses
2g
v
D
Lfh
2
L
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4.6 Minor and Major Losses
Minor losses are usually expressed in terms of loss coefficient
which is also called resistance coefficient where hL is the
additional irreversible head loss in the piping system caused by insertion of
the component and is defined as
g2/V
hK
2L
L
g
Ph LL
2
VKP
2
LL
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ghL
2
KP LL
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Minor losses are also expressed in terms of the equivalent length Lequiv
Where f is the friction factor and D is the diameter of the pipe that contains the
component.
4.6 Minor and Major Losses
Lequiv
2equiv
2
LL Kf
DL
g2
V
D
Lf
g2
VKh
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The head loss caused by the component is equivalent to the head loss
caused by a section of the pipe whose length is Lequiv. This is simply
accounted for additional length for the pipe.
Minor Head Loss
g2
VKh
2
LL8
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The total head losses therefore for a given pipe length with any of the
components (valves, elbows, etc) is the combination of the equation 7 and 8
i.e.
4.6 Minor and Major Losses
hhh minormajortotal ,L,L,L
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For constant diameter
2g
V
2g
V
D
Lfh
2
j2
i
i
iitotal
jj,L
i,L K 9
2g
V K
D
Lf h
2
Ltotal
,L 10
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4.6 Minor and Major Losses
Sharp-Edged Exit
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4.6 Minor and Major Losses
Loss Coefficient for sudden contraction
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4.6 Minor and Major Losses
Loss Coefficient for sudden expansion
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4.6 Minor and Major Losses
Loss Coefficient for Round Edge
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4.6 Minor and Major Losses
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4.6 Minor and Major Losses
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4.6 Minor and Major Losses
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4.6 Minor and Major Losses
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4.5 Minor and Major Losses
PIPE INLETS
Reentrant: KL = 0.80
T
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Example 4.4
Water at 10C flows from a large reservoir to a smaller one
through a 5-cm diameter cast iron piping system, as shown in
Figure below. Determine the elevation z1 for a flow rate of 6 L/s.
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FLUID MECHANICSLearning Outcomes