4b HH223 Bernoulli

8/13/2019 4b HH223 Bernoulli

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UNIVERSIDADNACIONALDEINGENIERA

FACULTADDEINGENIERACIVIL

DEPARTAMENTOACADMICODEHIDRULICAEHIDROLOGA

MECNICADEFLUIDOS

(HH 223K)

4b-Bernoulli.pptx


2/22

Las fuerzas que actuan sobre un elemento de fluido:

ECUACINDEBERNOULLI(1)


3/22

Como la aceleracion es:

para un flujo permanente, la segunda componente se

anula.

Del grfico, podemos afirmar que:

Reemplazando:

EC.EULER



4/22

Despejando y ordenando:

y se concluye que:

Es decir:

Esta es la ECUACION DE BERNOULLI



5/22

La ecuacin puede escribirse tambien como:



6/22

Altura(carga)piezomtrica

Altura(carga)total

Presinesttica(presinPdelacargadepresin)

Presindeestancamiento

CONCEPTOSCLAVE


7/22

The Bernoulli Equation (unit of L)

Atanytwopointsonastreamline:

P1/+V

1

2/2g+z1=P

2/+V

2

2/2g+z2

12


8/22

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)


9/22

StagnationPoints

Onanybodyinaflowingfluid,thereisastagnationpoint. Somefluidflows

overandsomeunderthebody.Thedividingline(thestagnationstreamline)terminatesatthestagnationpoint. TheVelocitydecreasesasthefluid

approachesthestagnationpoint. Thepressureatthestagnationpointisthepressureobtainedwhenaflowingfluidisdeceleratedtozerospeedbya

frictionlessprocess


10/22

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


11/22

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


12/22

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))


13/22

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>


14/22

TheEnergyLineandtheHydraulicGradeLine

LookingattheBernoulliequationagain:

P/+V2/2g +z=constantonastreamline Thisconstantiscalledthetotalhead(energy),H

Becauseenergyisassumedtobeconserved,atanypointalongthestreamline,thetotalheadisalwaysconstant

EachtermintheBernoulliequationisatypeofhead.

P/=PressureHead

V2/2g =VelocityHead

Z=elevationheadThesethreeheadssummedequalsH=totalenergy

Nextwewilllookatthisgraphically


15/22


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


16/22


Q

Z

P/

V2/2gEL

HGL

UnderstandingthegraphicalapproachofEnergyLineandtheHydraulicGradelineiskeytounderstandingwhatforcesaresupplyingtheenergythatwaterholds.

V2/2g

P/

Z

1

2

Point1:

Majorityofenergystored

inthewaterisinthePressureHead

Point2:

Majorityofenergystoredinthewaterisintheelevationhead

Ifthetubewassymmetrical,thenthevelocitywouldbeconstant,andtheHGLwouldbelevel


17/22


18/22

Problema Porunatuberahorizontalde20mmdedimetrocirculaun

fluidoconunavelocidadde3m/s.Enseguidaeltuboseestreche

aundimetrode10mm.Sielfluidoesaguacalcularladiferenciadealturasentredostubosverticalescolocadosinmediatamenteantesydespusdelestrechamiento.


19/22

Mide la presion total.

TUBODE

PITOT


20/22

EJEMPLO1


21/22

EJEMPLO2


22/22

EJEMPLO3

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4b HH223 Bernoulli