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StudyNotesonFlowthroughPipesforGATEFlowthroughPipesisaveryimportanttopicofFluidMechanicsfromGATEperspective.Thisisahigh-yieldingtopicsinceitcarriesahighweightageinGATE.Althoughthistopicappearstobetoughtomajorityofthecandidates,attainingafairdegreeofcompetenceinthistopiccanhelpyouboostyouroverallscoreinGATE.ThispostwillfamiliarizeyouwithallaspectsassociatedwithFlowthroughPipes.

FlowthroughPipes:AssociatedConcepts

I. Introduction

• HydraulicTurbine&Pumps

HydraulicMachinesrefertothosemachineswhichconverteithermechanicalenergyintohydraulicenergyorhydraulicenergyintomechanicalenergy.

• HeadHeadreferstotheenergycontentoffluidperunitweightoffluid.

HeadH=!"#$%'()*+,

-).+/#"01%2.3

Itismeasuredinmetreorcentimeter

• DynamicHeadItisalsocalledastheKineticHeadandistheratioofkineticenergytoweightofthefluid.

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• StaticHead

StaticHeadisdefinedastheratioofPotentialEnergyofthefluidtoitsweight.

• PressureHeadItisdefinedastheratioofpressureenergytoitsweight.

PressureHead= 45+

• PiezometricHead

Itimpliesthesumofpressureheadandpotentialhead.

PiezometicHead=56+h

II. FlowofIncompressibleFluidsinPipes

• LaminarFlow:LaminarFlowtakesplacewhenafluidflowsinparallel

layerswithnodisruptionbetweenthelayers.Themotionoftheparticlesofthefluidisveryorderlyinalaminarflow.

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InlaminarFlow,theReynoldsnumberRe<2000

• TurbulentFlow:Turbulentflowreferstoatypeoffluidflowwhereinthefluidundergoesirregularfluctuationsormixing.Thespeedofthefluidatanypointcontinuouslyundergoeschangesbothinmagnitudeaswellasdirectioninturbulentflow.Inturbulentflow,theReynoldsnumber>4000

• TurbulentFlowinCircularPipesTheheadlossinturbulentflowinacircularpipeisdenotedby: hf=2fLv2/D=Δp/ρ

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wherefdenotesthefrictionfactorrepresentedasf=tw/(ρv2/2)

wheretwrepresentsthewallshearstressThevalueofthefrictionfactordependsuponthefollowingparameters:

i. Velocity(v)ii. Densityoffluid(ρ)iii. PipeDiameter(D)iv. Viscosityoffluid(μ)v. AbsoluteRoughness(k)ofthepipe

• VelocityDistributioninTurbulentFlow

Anumberofsemi-theoreticalexpressionshavebeendevelopedfortheshearstressatthewallsofapipeofcircularcross-sectionThevelocityatanypointinthecross-sectionisdefinedbythefollowingexpressionwherein

ThisequationisalsoknownasthePrandtlone-seventhpowerlaw.Thissuggeststhatthevelocityisproportionalto1/7thpowerofthedistancefromthewalls.Inthisexpression,uxdenotesthevelocityatadistanceyfromthewalls,

uCLdenotesthevelocityatthecenterlineofpiperdenotestheradiusofthepipe

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• FlowthroughNon-circularpipesForturbulentflowinapipeofnon-circularcross-section,the‘hydraulicmeandiameter’canbeusedasasubstituteinplaceofthepipediameter.HydraulicMeanDiameterDH=4(HydraulicMeanRadiusrH)HydraulicMeanRadiusiscomputedbydividingtheflowcross-sectionalareabythewettedperimeter.i. ForCircularpipe,DH=Dii. IncaseofanannulusofouterdiaDoandinnerdiaDi

DH=Do-Di

iii. Incaseofaductofrectangularcross-sectionDabyDbDH=2DaDb/(Da+Db)

iv. Incaseofaductofsquarecross-sectionDa

v. DH=Da

• FlowthroughCurvedPipesIncasethepipeiscurved,thevelocitydistributionoverthesectionwillbealteredandthedirectionofflowoffluidwillkeeponchangingcontinuously.Thefrictionallossesaresomehowgreaterincomparisontoastraightpipeofthesamelength.StablelaminarflowpersistsathighervaluesofReynoldsnumberincurvedpipes.

III. EnergyLossesthroughPipes

• MajorEnergyLosses

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HeadLosshf=

Wherefisthecoefficientoffriction

ForLaminarflow,f=789)

ForTurbulentflow,frictionfactorisbasedontheMoodychart.

• MinorEnergyLossesi. LossofHeadduetosuddenenlargement

isdenotedby

ii. Lossofheadduetosuddencontractionisdenotedby

wherekdenotesadynamiclosscoefficient=0.375forCc=0.62

iii. LossofheadduetoObstructioninpipeisdenotedby

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whereAdenotestheareaofpipeAdenotestheareaofobstruction

iv. LossofheadattheEntranceofpipeisdenotedby

v. LossofHeadattheExitofpipe

isdenotedby

vi. LossofHeadduetobendinthepipeisdenotedby

wherekdependsonradiusofcurvature,angleofbendandpipediameter

vii. LossofHeadinvariousPipeFittingsisdenotedby

wherekdependsontypeofpipefittings

IV. EquivalentPipe

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Itreferstoapipeofuniformdiameterhavingequallossofheadanddischargetothatofacompoundpipecomprisingseveralpipesofdifferentlengthsanddiameters.Dupit’sequationisusedtodeterminethesizeoftheequivalentpipe.

V. PowerTransmittedthroughPipesPowertransmittedthroughpipeswillbemaximumwhen

SolvedExamplesfromGATE

Q1. Thevelocityprofileofafullydevelopedlaminarflowinastraightcircularpipe

isdenotedbytheexpression

where𝒅𝒑𝒅𝒙

denotesaconstant.

Thelaminarflowisillustratedbythefigurebelow:

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Computetheaveragevelocityofthefluidinthepipe?

A.

B.

C.

D.

CorrectAnswer:OptionASolution:

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Solvingweget,

Q=AreaxAverageVelocity

Q2.Neglectinglossesinthecylinderandassumingfullydevelopedlaminarviscousflow

throughouttheneedle,theDarcyfrictionfactorisgivenby𝟔𝟒𝑹𝒆whereRedenotesthe

Reynoldsnumber.Assumingtheviscosityofthewatertobe1.0x10-3Kg/ms,computetheforceisNewtonrequiredontheplunger?

A. 0.13B. 0.16C. 0.3D. 4.4

CorrectAnswer:OptionCExplanation:GivenViscosityofWater,v=1.0x10-3Kg/msReynoldsnumberisgivenby

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Re=1000

Now,DarcyFrictionfactorf=8A9)= 8A7BBB

=0.064HeadLossinNeedleisgivenby

=0.3265mofwater

ByapplyingBernoulli’sEquationatpoints1&2,

Sincez1=z2andP2=0

=499.95+3199.7=3699.65

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ForcerequiredonPlunger=P1xA1

=0.3N

Q3Waterisflowingthrougha1KmlongG.I.pipeat25°C.Thediameterofthepipeis200mmandtherateofflowis0.07m3/s.AssumingthevalueoftheDarcyfrictionfactortobe0.02andthedensityofwatertobe1000Kg/m3,ComputethepumpingpowerinKWrequiredtomaintaintheflow?

A. 1.8B. 17.4C. 20.5D. 41.0

CorrectAnswer:OptionBExplanation:

Headlossis

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=9.81x0.07x25.37=17.4KWQ4Themaximumvelocityofa1-dimensionalincompressiblefullydevelopedviscousflowis6m/sbetweentwoparallelplates.Computethemeanvelocityinm/softheflow?

A. 2B. 3C. 4D. 5

CorrectAnswer:OptionCExplanation:

Umax=6m/s

Umean=Umax/1.5

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Umean=87.D=4m/s

HopethatthispostwilldefinitelyfamiliarizeyouwithallaspectsassociatedwithFlowthroughPipes.