4b hh223 bernoulli
TRANSCRIPT
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UNIVERSIDADNACIONALDEINGENIERA
FACULTADDEINGENIERACIVIL
DEPARTAMENTOACADMICODEHIDRULICAEHIDROLOGA
MECNICADEFLUIDOS
(HH 223K)
4b-Bernoulli.pptx
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Las fuerzas que actuan sobre un elemento de fluido:
ECUACINDEBERNOULLI(1)
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Como la aceleracion es:
para un flujo permanente, la segunda componente se
anula.
Del grfico, podemos afirmar que:
Reemplazando:
EC.EULER
ECUACINDEBERNOULLI(2)
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Despejando y ordenando:
y se concluye que:
Es decir:
Esta es la ECUACION DE BERNOULLI
ECUACINDEBERNOULLI(3)
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La ecuacin puede escribirse tambien como:
ECUACINDEBERNOULLI(5)
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Altura(carga)piezomtrica
Altura(carga)total
Presinesttica(presinPdelacargadepresin)
Presindeestancamiento
CONCEPTOSCLAVE
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The Bernoulli Equation (unit of L)
Atanytwopointsonastreamline:
P1/+V
1
2/2g+z1=P
2/+V
2
2/2g+z2
12
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A Simple Bernoulli Example
V2Z
= air
Determine the difference in pressure between points 1 and 2
Assume a coordinate system fixed to the bike (from this system,
the bicycle is stationary, and the world moves past it). Therefore,
the air is moving at the speed of the bicycle. Thus, V2 = Velocity of
the Biker
Hint: Point 1 is called a stagnation point, because the air particle
along that streamline, when it hits the bikers face, has a zero
velocity (see next slide)
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StagnationPoints
Onanybodyinaflowingfluid,thereisastagnationpoint. Somefluidflows
overandsomeunderthebody.Thedividingline(thestagnationstreamline)terminatesatthestagnationpoint. TheVelocitydecreasesasthefluid
approachesthestagnationpoint. Thepressureatthestagnationpointisthepressureobtainedwhenaflowingfluidisdeceleratedtozerospeedbya
frictionlessprocess
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Apply Bernoulli from 1 to 2
V2Z
Point 1 = Point 2P1/air+ V12/2g + z1 = P2/air+ V22/2g + z2
Knowing the z1
= z2
and that V1
= 0, we can simplify
the equation
P1/air= P2/air+ V22/2g
P1 P2 = ( V22/2g ) air
= air
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Bernoulli Example Problem: Free JetsWhatistheFlowRateatpoint2? Whatisthevelocityatpoint3?
1
2
3
H2O
Part1:
ApplyBernoulliseqn betweenpoints1and2
P1/H2O+V12/2g +h=P2/H20+V2
2/2g +0
simplifiesto
h=V22/2g solvingforV V =
(2gh)
Q=VA or Q=A2(2gh)0A2
GivensandAssumptions: Becausethetankissolarge,weassumeV1=0(Volout
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1
2
3
H2O
Z=0A2
Bernoulli Example Problem: Free Jets
Part2:FindV3?
ApplyBernoulliseqfrompt1topt3P1/H2O+V1
2/2g+h=P3/H20+V32/2g H
Simplifyto h+H=V32/2g
SolvingforV V3= (2g(h+H))
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TheContinuityEquationWhydoesahosewithanozzleshootwaterfurther?
ConservationofMass: Inaconfinedsystem,allofthemassthatentersthesystem,mustalsoexitthesystematthe
sametime.
Flowrate=Q=AreaxVelocity
1A1V1(massinflowrate)=2A2V2(massoutflowrate)
Ifthefluidatbothpointsisthesame,thenthedensitydropsout,
andyougetthecontinuityequation: A1V1=A2V2
Therefore IfA2 V1
Thus,waterexitinganozzlehasahighervelocity
Q1=A1V1
A1
V1>
Q2=A2V2
A1V1=A2V2
A2 V2>
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TheEnergyLineandtheHydraulicGradeLine
LookingattheBernoulliequationagain:
P/+V2/2g +z=constantonastreamline Thisconstantiscalledthetotalhead(energy),H
Becauseenergyisassumedtobeconserved,atanypointalongthestreamline,thetotalheadisalwaysconstant
EachtermintheBernoulliequationisatypeofhead.
P/=PressureHead
V2/2g =VelocityHead
Z=elevationheadThesethreeheadssummedequalsH=totalenergy
Nextwewilllookatthisgraphically
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TheEnergyLineandtheHydraulicGradeLine
Q
Measuresthestaticpressure
Pitot measuresthetotalhead
1
Z
P/
V2/2g EL
HGL
2
1:StaticPressureTap
Measuresthesumofthe
elevationheadandthepressureHead.
2:Pitot Tube
MeasurestheTotalHead
EL:EnergyLine
TotalHeadalongasystem
HGL:HydraulicGradeline
Sumoftheelevationandthepressureheadsalonga
system
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TheEnergyLineandtheHydraulicGradeLine
Q
Z
P/
V2/2gEL
HGL
UnderstandingthegraphicalapproachofEnergyLineandtheHydraulicGradelineiskeytounderstandingwhatforcesaresupplyingtheenergythatwaterholds.
V2/2g
P/
Z
1
2
Point1:
Majorityofenergystored
inthewaterisinthePressureHead
Point2:
Majorityofenergystoredinthewaterisintheelevationhead
Ifthetubewassymmetrical,thenthevelocitywouldbeconstant,andtheHGLwouldbelevel
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Problema Porunatuberahorizontalde20mmdedimetrocirculaun
fluidoconunavelocidadde3m/s.Enseguidaeltuboseestreche
aundimetrode10mm.Sielfluidoesaguacalcularladiferenciadealturasentredostubosverticalescolocadosinmediatamenteantesydespusdelestrechamiento.
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Mide la presion total.
TUBODE
PITOT
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EJEMPLO1
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EJEMPLO2
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EJEMPLO3