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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1
Reynolds—Transport Theorem (RTT
! system is a "uantity o# matter o# #i$ed identity% Nomass can cross a system boundary.
! control volume is a re&ion in space chosen #or study%Mass can cross a control sur#ace%
The #undamental conser'ation laws (conser'ation o#
mass ener&y and momentum apply directly to systems%)owe'er in most #luid mechanics pro*lems control'olume analysis is pre#erred o'er system analysis
There#ore we need to trans#orm the conser'ation laws#rom a system to a control 'olume% This is accomplished
with the Reynolds transport theorem (RTT%
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +
!'era&e ,elocity and ,olume Flow Rate
1
c
avg n c
c A
V V dA A= ∫
avg cm V A ρ =&
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 3
Mass and ,olume Flow Rates
c c
n c
A A
m m V dAδ ρ = =∫ ∫ &
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 4
Reynolds—Transport Theorem (RTT
-eneral RTT non#i$ed C, (inte&ral analysis:
( ) sys
CV CS
dBb dV bV ndA
dt t ρ ρ
∂= +
∂∫ ∫ r rg
Mass Momentum Energy Angularmomentum
. E$tensi'e properties m E* /ntensi'e properties 1 e
mV V
H ( )r V ×
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 0
Conser'ation o# Mass rinciple
( ) 0CV CS
d dV V n dAdt ρ ρ + =∫ ∫
r rg
( )
sys
CV CS
dB
b d
V bV ndAdt t ρ ρ
∂
= +∂∫ ∫
r r
g
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 2
teady—Flow rocesses
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Chapter 4: Fluid KinematicsME33 : Fluid Flow
5ewton6s 7aws
Newton’s laws are relations between motions of bodiesand the forces acting on them.
First law: a body at rest remains at rest, and a body in motion
remains in motion at the same velocity in a straight path whenthe net force acting on it is zero.
Second law: the acceleration of a body is proportional to thenet force acting on it and is inversely proportional to its mass.
Third law: when a *ody e$erts a #orce on a second *ody the
second *ody e$erts an e"ual and opposite #orce on the #irst%
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 8
Choosing a Control Volume
Selection of CV can either simplify orcomplicate analysis.
Fixed, moving, and deforming controlvolumes: for mass flow calculation userelative velocity ;
but use absolute velocity for Newton’s Law!
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 9
Forces Acting on a CV
Forces acting on CV consist ofbody forcesthat actthroughout the entire body of the CV (such as gravity,electric, and magnetic forces) andsurface forcesthatact on the control surface (such as pressure and viscousforces, and reaction forces at points of contact).
•Body forces act on eachvolumetric portiondVof the CV.
•Surface forces act on each
portiondAof the CS.
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1
.ody Forces
The most common *ody #orceis &ra'ity which e$erts a
downward #orce on e'ery
di##erential element o# the C,
The di##erent *ody #orce
Typical con'ention is that
acts in the ne&ati'e z ;direction
Total *ody #orce actin& on C,
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 11
ur#ace Forces
ur#ace #orces are not as simple toanaly<e since they include *oth normal
and tan&ential components
=ia&onal components σ xx , σ yy , σ zz are
called normal stresses and are due to
pressure and 'iscous stresses>##;dia&onal components σ xy , σ xz , etc%
are called shear stresses and are due
solely to 'iscous stresses
Total sur#ace #orce actin& on C
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1+
7inear Momentum E"uation
5ewton6s second law #or a system:
?se RTT with b = V and B = mV:
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 13
pecial Cases
teady Flow !'era&e 'elocities
!ppro$imate momentum #low rate
To account #or error use momentum;#lu$
correction #actor β
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 14
ro*lem
For the el*ow duct !E3 oil at +@C (speci#ic wei&ht is 8+ 5Am3enters section 1 at 30 5As where the #low is laminar and e$its at
section + where trhe #low is tur*ulent: u1B,ma$1(1;(rAR1+
u+B,ma$+(1;(rAR+(1A% !ssumin& steady incompressi*le #low
compute the #orce and its direction o# the oil on the el*ow due to
momentum chan&e only (no pressure chan&es or #riction e##ects%=1B1 cmD =+B2 cmD an&leB3@%
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 10
ro*lem
-ra'el is dumped #rom a hopper at a rate o# 20 5As onto a mo'in&*elt as in Fi&% The &ra'el then passes o## the end o# the *elt% The
dri'e wheels are 8 cm in diameter and rotate clocwise at 10
rAmin% 5e&lectin& system #riction and air dra& estimate the power
re"uired to dri'e this *elt%
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 12
!n&ular Momentum
Motion of a rigid body can be considered to be thecombination of
the translational motion of its center of mass (U x, U y, U z)
the rotational motion about its center of mass (ω x,ω y,ω z)
Translational motion can be analyzed with linearmomentum equation.
Rotational motion is analyzed with angular momentumequation.
Together, the body motion can be described as a 6–degree–of–freedom (6DOF) system.
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1
Re'iew o# Rotational Motion
!n&ular 'elocity ω is thean&ular distance θ tra'eled per unit time and
an&ular acceleration α is
the rate o# chan&e o#an&ular 'elocity%
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 18
Re'iew o# !n&ular Momentum
Moment o# a #orce:Moment o# momentum:
For a system:
There#ore the an&ular momentum e"uation can
*e written as:To deri'e an&ular momentum #or a C, use RTT
with and
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 19
!n&ular Momentum E"uation #or a C,
-eneral #orm
!ppro$imate #orm usin& a'era&e
properties at inlets and outlets
teady #low
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +
ro*lem
The hori<ontal lawn sprinler in Fi& has a water #low rateo# 4% &alAmin introduced 'ertically throu&h the center%
Estimate (a the retardin& tor"ue re"uired to eep the
arms #rom rotatin& and (* the rotation rate (rAmin i#
there is no retardin& tor"ue%
d B in
R B 2 in
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +1
Mechanical Ener&y
Mechanical ener&y chan&e o# a #luid durin&incompressi*le #low
( )2 2
2 1 2 1
2 1
2
mech
P P V V e g z z
ρ
− −∆ = + + −
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Chapter 4: Fluid KinematicsME33 : Fluid Flow ++
-eneral Ener&y E"uation
Recall &eneral RTT
G=eri'eH ener&y e"uation usin& B=E and b=e
.rea power into rate o# sha#t and pressure wor
( ) sys
r CV CS
dB d bdV b V n dA
dt dt ρ ρ = +∫ ∫
r rg
( ), ,
sys
net in net in r CV CS
dE d Q W edV e V n dA
dt dt ρ ρ = + = +∫ ∫
r r& & g
( ), , , , , , ,net in shaft net in pressure net in shaft net inW W W W P V n dA= + = − ∫ r r& & & & g
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +3
-eneral Ener&y E"uation
Ihere does e$pression #or pressure worcome #romJ
Ihen piston mo'es down ds under thein#luence o# F=PA the wor done on thesystem is δ W boundary =PAds.
/# we di'ide *oth sides *y dt we ha'e
For &enerali<ed control 'olumes:
5ote si&n con'entions:
is outward pointin& normal
5e&ati'e si&n ensures that wor done ispositi'e when is done on the system%
pressure boundary piston
dsW W PA PAV
dt δ δ = = =& &
( ) pressure nW PdAV PdA V nδ = − = − ×
r&
n
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +4
-eneral Ener&y E"uation
Mo'in& inte&ral #or rate o# pressure worto R) o# ener&y e"uation results in:
Recall that P/ ρ is the flow work, which isthe wor associated with pushin& a #luidinto or out o# a C, per unit mass%
( ), , ,net in shaft net in r
CV CS
d P
Q W edV e e V n dAdt ρ ρ
+ = + + × ÷ ∫ ∫
r r&
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +0
-eneral Ener&y E"uation
!s with the mass e"uation practical analysis iso#ten #acilitated as a'era&es across inlets and
e$its
ince e=u+ke+e = u+V ! /!+"z
( )
, , ,
C
net in shaft net in
out inCV
c
A
d P P Q W edV m e m e
dt
m V n dA
ρ
ρ ρ
ρ
+ = + + − +
÷ ÷ = ×
∑ ∑∫
∫
& & &
r r
2 2
, , ,2 2
net in shaft net in
out inCV
d P V P V Q W edV m u gz m u gz
dt ρ
ρ ρ
+ = + + + + − + + + ÷ ÷
∑ ∑∫ & & &
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +2
Ener&y !nalysis o# teady Flows
For steady #low time rate o# chan&e o# the
ener&y content o# the C, is <ero%This e"uation states: t#e net rate o$ ener"ytrans$er to a %V by #eat and &ork trans$ersdur'n" steady $(o& 's e)ua( to t#e d'$$erence
bet&een t#e rates o$ out"o'n" and incomin&ener&y #lows with mass%
2 2
, , ,2 2
net in shaft net in
out in
V V Q W m h gz m h gz + = + + − + + ÷ ÷ ∑ ∑& &
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +
Ener&y !nalysis o# teady Flows
For single-streamdevices mass #low rate
is constant%
( )
( )
2 2
2 1, , , 2 1 2 1
2 2
1 1 2 2
, , 1 2 2 1 ,
1 2
2 2
1 1 2 2
1 2 ,
1 2
2
2 2
2 2
net in shaft net in
shaft net in net in
pump turbine mech loss
V V ! h h g z z
P V P V ! gz gz u u
P V P V gz ! gz ! e
ρ ρ
ρ ρ
−+ = − + + −
+ + + = + + + − −
+ + + = + + + +
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +8
Ener&y !nalysis o# teady Flows
=i'ide *y " to &et each term in units o# len&th
Ma&nitude o# each term is now e$pressed as an
e"ui'alent column hei&ht o# #luid i%e% *ead
2 2
1 1 2 2
1 2
1 22 2
pump turbine "
P V P V z h z h h
g g g g ρ ρ + + + = + + + +
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Chapter 4: Fluid KinematicsME33 : Fluid Flow +9
The .ernoulli E"uation
/# we ne&lect pipin& losses and ha'e a system withoutpumps or tur*ines
This is the Bernoulli equation
/t can also *e deri'ed usin& 5ewtons second law o#motion (see Cen&el te$t p% 18%
3 terms correspond to: tatic dynamic and hydrostatichead (or pressure%
2 2
1 1 2 2
1 21 22 2
P V P V z z
g g g g ρ ρ + + = + +
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 3
)-7 and E-7
/t is o#ten con'enient
to plot mechanical
ener&y &raphically
usin& hei&hts%
)ydraulic -rade 7ine
Ener&y -rade 7ine(or total ener&y
P H#" z
g ρ = +
2
2
P V E#" z
g g ρ = + +
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 31
The .ernoulli E"uation
The Bernoulli equation is an arox'mate re(at'on
bet&een ressure,
e(oc'ty and e(eat'on
and 's a('d 'n re"'ons o$
steady, 'ncomress'b(e
$(o& &#ere net $r'ct'ona(
$orces are ne"('"'b(e.
E"uation is use#ul in #low
re&ions outside o#*oundary layers and
waes%
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 3+
The .ernoulli E"uation
7imitations on the use o# the .ernoulli E"uationteady #low: d/dt B
Frictionless #low
5o sha#t wor: wpumpBwtur*ineB
/ncompressi*le #low: ρ B constant
5o heat trans#er: )net,'nB
!pplied alon& a streamline (e$cept #or irrotational
#low
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 33
ro*lem
Ihen the pump in Fi&% draws ++ m3Ah o# water at +@C #rom the reser'oir the total
#riction head loss is 0 m% The #low dischar&es throu&h a no<<le to the atmosphere%
Estimate the pump power in I deli'ered to the water%
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 34
ro*lem
.ernoulli6s 138 treatise *ydrodynam'ca contains many e$cellent setches o# #low
patterns% >ne howe'er redrawn here as Fi&% seems physically misleadin&% Ihat is
wron& with the drawin&J
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Chapter 4: Fluid KinematicsME33 : Fluid Flow 30
ro*lem
In the spillway fow o Fig., the fow is assumed uniorm
and hydrostatic at sections 1 and 2. I losses areneglected, compute (a) V 2 and (b) the orce per unit widtho the water on the spillway.
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