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INTERNAL FLOWS Fluid Mechanics II
General Concept of Flows in Pipe 2
p As a uniform flow enters a pipe, the velocity at the pipe walls must decrease to zero (no-slip boundary condition). Continuity indicates that the velocity at the center must increase.
p Thus, the velocity profile is changing continuously from the pipe entrance until it reaches a fully developed condition. This distance, L, is called the entrance length.
Dr. M. Khosravy
p For fully developed flows (x>>L), flows become parallel, , the mean pressure remains constant over the pipe cross-section
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v = (u(y),0,0)
General Concept of Flows in Pipe
Dr. M. Khosravy
General Concept of Flows in Pipe 4
l Flows in a long pipe (far away from pipe entrance and exit region, x>>L) are the limit results of boundary layer flows. There are two types of pipe flows: laminar and turbulent
Dr. M. Khosravy
l CriJcal Reynolds number (Recr) for flow in a round pipe Re < 2300 : laminar 2300 ≤ Re ≤ 4000 : transiJonal Re > 4000 : turbulent
l Note that these values are approximate.
l For a given applicaJon, Recr depends upon
l Pipe roughness l VibraJons l Upstream fluctuaJons, disturbances
(valves, elbows, etc. that may disturb the flow)
General Concept of Flows in Pipe
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Dr. M. Khosravy
General Concept of Flows in Pipe 6
l Whether the flow is laminar or turbulent depends on the Reynolds number, where Um is the cross-‐secJonal mean velocity defined by
l TransiJon from laminar to turbulent for flows in circular pipe of diameter D occur at Re=2300
Um =1A
udAA!!
Dr. M. Khosravy
Um Um
General Concept of Flows in Pipe
l For pipes with variable diameter, m is sJll the same due to conservaJon of mass, but V1 ≠ V2
D2
V2
2
1
V1
D1
m m
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Dr. M. Khosravy
General Concept of Flows in Pipe 8
l When pipe flow is turbulent. The velocity is unsteadily random (changing randomly with Jme), the flow is characterized by the mean (Jme-‐averaged) velocity defined as:
l Due to turbulent mixing, the velocity profile of turbulent pipe flow is more uniform then that of laminar flow.
v (y) = limT!"
12T
v(y, t)dt#T
T
$
Dr. M. Khosravy
Dr. M. Khosravy
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l Hence, the mean velocity gradient at the wall for turbulent flow is larger than laminar flow.
l The wall shear stress, ,is a funcJon of the velocity gradient. The greater the change in with respect to y at the wall, the higher is the wall shear stress. Therefore, the wall shear stress and the fricJonal losses are higher in turbulent flow.
wτu
General Concept of Flows in Pipe
Poiseuille Flow
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l Consider the steady, fully developed laminar flow in a straight pipe of circular cross secJon with constant diameter, D.
l The coordinate is chosen such that x is along the pipe and y is in the radius direcJon with the origin at the center of the pipe.
τD
b
y
x1p 2p
Dr. M. Khosravy
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l For a control volume of a cylinder near the pipe center, the balance of momentum in integral form in x-‐direcJon requires that the pressure force,
acJng on the faces of the cylinder be equal to the shear stress acJng on the circumferenJal area, hence
l In accordance with the law of fricJon (Newtonian fluid), have:
(p1 ! p2 )! y2
2! yb"
! = (p1 ! p2 )" y2b
! (y) = !µ dudy since u decreases with increasing y
Poiseuille Flow
l Therefore:
when is constant (negaJve)
l Upon integraJon:
The constant of integraJon, C, is obtained from the condiJon of no-‐slip at the wall. So, u=0 at y=R=D/2, there fore C=R2/4 and finally:
u(y) = ! p1 ! p24µb
R2 ! y2( ) = ! 1µdpdx
C ! y2
4"#$
%&'
dudy
= (p2 ! p1)!µb
y2= !µb
dpdx
y2
u(y) = ! 14µ
dpdx
R2 ! y2( )
dudy
= (p2 ! p1)b
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Dr. M. Khosravy
Poiseuille Flow
Poiseuille Flow
Dr. M. Khosravy
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l The velocity distribuJon is parabolic over the radius, and the maximum velocity on the pipe axis becomes:
l Therefore,
l The volume flow rate is:
um = ! 14µ
dpdxR2
uum
=1! y2
R2
Q = u(y)2!ydy" = #1
4µ"dpdx
R2 # y 2( )2!ydy
Q = #!8µ
dpdxR4 =
12um!R
2
Poiseuille Flow
Dr. M. Khosravy
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l The flow rate is proporJonal to the first power of the pressure gradient and to the fourth power of the radius of the pipe.
l Define mean velocity as
l Therefore,
l This soluJon occurs in pracJce as long as,
Um = Q!R2
Um = 12um = 1
8µ! dpdx
"#$
%&' R
2
Re =UmDv< 2300
! dpdx
= 8µR2UmHence,
Poiseuille Flow
Dr. M. Khosravy
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l The relaJon between the negaJve pressure gradient and the mean velocity of the flow is represented in engineering applicaJon by introducing a resistance coefficient of pipe flow, f.
l This coefficient is a non-‐dimensional negaJve pressure gradient using the dynamic head as pressure scale and the pipe diameter as length scale, i.e.,
l Introducing the above expression for (-‐dp/dx), so,
f = 2D!Um
28µUm
R2= 32µ!Um
2R
Ref 64=
! dpdx
= fD12!Um
2
Poiseuille Flow
Dr. M. Khosravy
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l At the wall,
l So,
l As a result, the wall fricJon coefficient is:
!w = !µ dudy y=R
= !µ dpdx2y4µ y=R
= ! R2dpdx
Cf =!w12"Um
2= f4= 16Re
2!wR
= ! dpdx
= fD12"Um
2
Head Loss in Pipe
Dr. M. Khosravy
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l For flows in pipes, the total energy per unit of mass is
given by where the correcJon factor is
defined as,
with being the mass flow rate and A is the cross secJonal area.
p!+ "Um
2
2+ gy
!"#
$%&
! = !u2"udA
A"!mUm
2
!m = !Q
α
Head Loss in Pipe
Dr. M. Khosravy
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l So the total head loss between secJon 1 and 2 of pipes is:
l hl=head loss due to fricJonal effects in fully developed flow in constant area conduits
l hlm=minor losses due to entrances, fi`ngs, area changes, etcs.
hlt =p1!
+"1Um1
2
2+ gy1
#
$%%
&
'(()
p2!
+"2Um2
2
2+ gy2
#
$%%
&
'((
hlt = hl + hlm
Head Loss in Pipe
Dr. M. Khosravy
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l So, for a fully developed flow through a constant-‐area pipe,
l And if y1=y2,
hl =p1 ! p2!
+ g(y1 ! y2 )
hl =p1 ! p2!
= "p!
Head Loss in Pipe
Dr. M. Khosravy
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l For laminar flow,
l Hence
! dpdx
= "pL
= ! 32µUm
D2
hl =!p!
= 32LµUm
D2 = 64µ!UmD
LDUm2
2= 64Re
LDUm2
2
hl = f LDUm2
2
Turbulent Pipe Flow
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l For turbulent flows’ we cannot evaluate the pressure drop analyJcally. We must use experimental data and dimensional analysis.
l In fully developed turbulent pipe flow, the pressure drop, , due to fricJon in a horizontal constant-‐area pipe is know to depend on: l Pipe diameter, D l Pipe length, L l Pipe roughness, e l Average flow velocity, Um l Fluid density, l Fluid viscosity,
pΔ
ρµ
Turbulent Pipe Flow
Dr. M. Khosravy
22
l Therefore,
l Dimensional analysis,
l Experiments show that the non-‐dimensional head loss is directly proporJonal to L/D, hence
!p!Um
2 ="1µ
!U mD, LD, eD
"#$
%&'
hl =!p!
( hl!Um
2 ="1 Re,LD, eD
"#$
%&'
!p = !p D,L,e,Um,!,µ( )
hlUm
2 / 2= LD!2 Re, e
D!"#
$%&
Turbulent Pipe Flow
Dr. M. Khosravy
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l Defining the fricJon factor as, , hence
where f is determined experimentally.
l The experimental result are usually ploded in a chart called Moody Diagram.
f =!2 Re,eD
!"#
$%&
hl = f LD
!"#
$%&Um2
2!"#
$%&
Moody Diagram
Dr. M. Khosravy
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Turbulent Pipe Flow
Dr. M. Khosravy
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l In order to solve the pipe flow problems numerically, a mathemaJcal formulaJon is required for the fricJon factor, f, in terms of the Reynolds number and the relaJve roughness.
l The most widely used formula for the fricJon factor is that due to Colebrook,
l This an implicit equaJon, so iteraJon procedure is needed to determine.
1f= !2 log e / D
3.7+ 2.51Re f
"
#$%
&'
Turbulent Pipe Flow
Dr. M. Khosravy
26
l Miller suggested to use for the iniJal esJmate,
l That produces results within 1% in a single iteraJon
fo = 0.25 loge / D3.7
+ 2.51Re0.9
!"#
$%&
'()
*+,
-2
Minor Loss
Dr. M. Khosravy
27
l The minor head loss may be expressed as,
where the loss coefficient, K, must be determined experimentally for each case.
l Minor head loss may be expressed as
where Le is an equivalent length of straight pipe
hlm = f LeD
!"#
$%&Um
2
2!"#
$%&
hlm = KU m2
2
Minor Loss
Dr. M. Khosravy
28
p Source of minor loss:
1. Inlets & Outlets 2. Enlargements & Contractions 3. Valves & Fittings 4. Pipe Bends
Minor Losses
l Total head loss in a system is comprised of major losses (in the pipe secJons) and the minor losses (in the components)
l If the piping system has constant diameter
i pipe sections
j components
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Dr. M. Khosravy
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Dr. M. Khosravy
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Dr. M. Khosravy
p Pipe flow results sometimes can be used for non-circular ducts or open channel flows to estimate the head loss
p Use Hydraulic Diameter,
A - Cross section area; P - Wetted perimeter p For a circular duct,
p For rectangular duct,
where Ar =b/a is the geometric aspect ratio
Non-Circular Ducts
Dr. M. Khosravy
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PADh4=
DDDDh ==ππ 4/4 2
)1(24)(24AraAr
baabDh +=+
=
Non-Circular Ducts
Dr. M. Khosravy
33
p Effect of Aspect Ratio (b/a): n For square ducts:
n For wide rectangular ducts with b>>a:
Thus, flows behave like channel flows n However, pipe flow results can be used with good
accuracy only when:
a=b Ar=1 Dh=a
Ar Dh2a ∞
1/3<Ar<3
b
a
ba
b
ab
a
Piping Networks and Pump SelecJon l Two general types of networks
l Pipes in series l Volume flow rate is constant l Head loss is the summaJon of parts
l Pipes in parallel l Volume flow rate is the sum of the components
l Pressure loss across all branches is the same
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Dr. M. Khosravy
Piping Networks and Pump SelecJon
l For parallel pipes, perform CV analysis between points A and B
l Since Δp is the same for all branches, head loss in all branches is the same
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Dr. M. Khosravy
Piping Networks and Pump SelecJon l Head loss relaJonship between branches allows the following raJos to be developed
l Real pipe systems result in a system of non-‐linear equaJons. Very easy to solve with EES!
l Note: the analogy with electrical circuits should be obvious l Flow flow rate (VA) : current (I) l Pressure gradient (Δp) : electrical potenJal (V) l Head loss (hL): resistance (R), however hL is very nonlinear
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Dr. M. Khosravy
Piping Networks and Pump SelecJon
l When a piping system involves pumps and/or turbines, pump and turbine head must be included in the energy equaJon
l The useful head of the pump (hpump,u) or the head extracted by the turbine (hturbine,e), are funcJons of volume flow rate, i.e., they are not constants.
l OperaJng point of system is where the system is in balance, e.g., where pump head is equal to the head losses.
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Dr. M. Khosravy
Pump and systems curves l Supply curve for hpump,u: determine experimentally by manufacturer. When using EES, it is easy to build in funcJonal relaJonship for hpump,u.
l System curve determined from analysis of fluid dynamics equaJons
l Ope r a J n g p o i n t i s t h e intersecJon of supply and demand curves
l If peak efficiency is far from operaJng point, pump is wrong for that applicaJon.
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Dr. M. Khosravy
Dr. M. Khosravy
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Dr. M. Khosravy
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