University Press of Florida

Basics of Fluid Mechanics

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Title:
Basics of Fluid Mechanics
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Book
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en-US
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Bar-Meir, Genick

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Subjects / Keywords:
Mathematics, Statistics, Science,Technology, Thermodynamics, Review of Mechanics. Statics, Multi-Phase Flow, OGT+ ISBN: 9781616100148 and 9781616100940
Mechanics (Physics)
Science / Engineering, Science / Physics

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Abstract:
This book describes the fundamentals of fluid mechanics phenomena for engineers and others. This book is designed to replace all introductory textbook(s) or instructor's notes for the fluid mechanics in undergraduate classes for engineering/science students but also for technical people. It is hoped that the book could be used as a reference book for people who have at least some basics knowledge of science areas such as calculus, physics, etc. This version is a PDF document. The website http://www.potto.org/FM/fluidMechanics.pdf contains the book broken into sections, and also has LaTeX resources.
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Expositive
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Community College, Higher Education
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http://www.ogtp-cart.com/product.aspx?ISBN=9781616100148
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Adobe Reader
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Genick Bar–Meir, Ph. D.
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Textbook
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barmeir@gmail.com
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http://florida.theorangegrove.org/og/file/3af22a83-06cb-53c6-c1d4-010dadbe5e17/1/fluidMechanics.pdf

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University of Florida
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University Press of Florida
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license. Site feedback please mail to: barmeir at gmail.com
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Version(0.1.6August11,2008)

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xi ...................... xv 1.APPLICABILITYANDDEFINITIONS ................ xvi 2.VERBATIMCOPYING ........................ xvii 3.COPYINGINQUANTITY ....................... xviii 4.MODIFICATIONS ........................... xviii 5.COMBININGDOCUMENTS ..................... xx 6.COLLECTIONSOFDOCUMENTS ................. xxi 7.AGGREGATIONWITHINDEPENDENTWORKS ......... xxi 8.TRANSLATION ............................ xxi 9.TERMINATION ............................ xxi 10.FUTUREREVISIONSOFTHISLICENSE ............. xxii ADDENDUM:HowtousethisLicenseforyourdocuments ...... xxii Howtocontributetothisbook ........................ xxiii Credits ..................................... xxiii Stevenfromartofproblemsolving.com ................. xxiii DanOlsen ................................ xxiv RichardHackbarth ............................ xxiv TousherYang ............................... xxiv Yournamehere ............................. xxiv Typocorrectionsandotherminorcontributions ........... xxiv Version0.1.8August6,2008 ......................... xxxiii pages189size2.6M ........................... xxxiii Version0.1April22,2008 ........................... xxxiii pages151size1.3M ........................... xxxiii Properties ................................. xxxix i

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........................... xxxix 1 ........................ 1 1.2BriefHistory ................................ 3 1.3KindsofFluids .............................. 5 1.4ShearStress ............................... 6 1.5Viscosity .................................. 9 1.5.1General .............................. 9 1.5.2NonNewtonianFluids ...................... 10 1.5.3KinematicViscosity ....................... 11 1.5.4EstimationofTheViscosity ................... 12 1.5.5BulkModulus ........................... 19 1.6SurfaceTension ............................. 22 1.6.1WettingofSurfaces ....................... 24 33 ............................. 33 41 .............................. 41 3.1.1CenteroftheMass ........................ 41 3.1.2CenterofArea .......................... 42 3.2MomentofInertia ............................. 43 3.2.1MomentofInertiaforMass ................... 43 3.2.2MomentofInertiaforArea .................... 44 3.2.3ExamplesofMomentofInertia ................. 46 3.2.4ProductofInertia ......................... 48 3.2.5PrincipalAxesofInertia ..................... 50 3.3Newton'sLawsofMotion ........................ 50 3.4AngularMomentumandTorque ..................... 51 3.4.1Tablesofgeometries ...................... 52 55 ................................ 55 4.2TheHydrostaticEquation ........................ 55 4.3PressureandDensityinaGravitationalField ............. 57 4.3.1ConstantDensityinGravitationalField ............. 57 4.3.2PressureMeasurement ..................... 59 4.3.3VaryingDensityinaGravityField ................ 61 4.3.4ThePressureEffectsBecauseTemperatureVariations .... 65 4.3.5GravityVariationsEffectsonPressureandDensity ...... 69 4.3.6LiquidPhase ........................... 71 4.4FluidinaAcceleratedSystem ...................... 72 4.4.1FluidinaLinearlyAcceleratedSystem ............. 72

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4.4.2AngularAccelerationSystems:ConstantDensity ....... 74 4.5FluidForcesonSurfaces ........................ 75 4.5.1FluidForcesonStraightSurfaces ................ 75 4.5.2ForceonCurvedSurfaces .................... 85 4.6BuoyancyandStability .......................... 92 4.6.1Stability .............................. 98 4.6.2SurfaceTension ......................... 107 4.7RayleighTaylorInstability ........................ 108 113 ................................ 113 5.2History ................................... 113 5.3WhattoExpectFromThisChapter ................... 114 5.4KindofMulti-PhaseFlow ........................ 115 5.5ClassicationofLiquid-LiquidFlowRegimes ............. 116 5.5.1CoCurrentFlow ......................... 117 5.6MultiPhaseFlowVariablesDenitions ................. 122 5.6.1MultiPhaseAveragedVariablesDenitions .......... 122 5.7HomogeneousModels .......................... 125 5.7.1PressureLossComponents ................... 126 5.7.2LockhartMartinelliModel .................... 128 5.8SolidLiquidFlow ............................. 129 5.8.1SolidParticleswithHeavierDensityS>L 130 5.8.2SolidWithLighterDensityS
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2 1.2Densityasafunctionofthesizeofsample. .............. 6 1.3Schematicstodescribetheshearstressinuidmechanics. ..... 6 1.4Thedeformationofuidduetoshearstressasprogressionoftime. 7 1.5thedifferenceofpoweruids ...................... 9 1.6Nitrogen(left)andArgon(right)viscosityasafunctionofthetem-peratureandpressureafterLemmonandJacobsen. ......... 10 1.7Theshearstressasafunctionoftheshearrate ............ 11 1.8Airviscosityasafunctionofthetemperature. ............. 12 1.9Waterviscosityasafunctiontemperature. ............... 12 1.10Liquidmetalsviscosityasafunctionofthetemperature. ....... 13 1.11Reducedviscosityasfunctionofthereducedtemperature. ..... 17 1.12Reducedviscosityasfunctionofthereducedtemperature. ..... 18 1.13SurfaceTensioncontrolvolumeanalysis. ............... 21 1.14ForcesinContactangle. ......................... 24 1.15Descriptionofwettingandnonwettinguids. ............. 24 1.16Descriptionofliquidsurface. ....................... 26 1.17Theraisingheightasafunctionoftheradii. .............. 29 1.18Theraisingheightasafunctionoftheradius. ............. 29 3.1Descriptionofhowthecenterofmassiscalculated. ......... 42 3.2Thinbodycenterofmass/areaschematic. ............... 42 3.3Theschematicthatexplainsthesummationofmomentofinertia. .. 44 3.4Theschematictoexplainthesummationofmomentofinertia. .... 45 3.5Cylinderwiththeelementforcalculationmomentofinertia. ..... 45 3.6Descriptionofrectangularinxyplaneforcalculationofmomentofinertia. ................................... 46 v

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.................... 47 3.8Theratioofthemomentofinertiaoftwo-dimensionaltothreedimensional. 47 3.9Descriptionofparabolaforcalculationofmomentofinertiaandcen-terofarea. ................................ 48 3.10Productofinertiafortriangle. ...................... 49 4.1Descriptionofauidelementinacceleratedsystemunderbodyforces. 55 4.2Pressurelinesastaticuidwithaconstantdensity. .......... 58 4.3Aschematictoexplainthemeasureoftheatmosphericpressure. .. 58 4.4SchematicofgasmeasurementutilizingtheUtube. ........ 59 4.5Schematicofsensitivemeasurementdevice. ............. 60 4.6Hydrostaticpressurewhenthereiscompressibilityintheliquidphase. 64 4.7Twoadjoinlayersforstabilityanalysis. ................. 67 4.8Thevaryinggravityeffectsondensityandpressure. ......... 69 4.9Theeffectivegravityisforacceleratedcart. .............. 73 4.10Acartslideoninclinedplane ...................... 73 4.11Forcesdiagramofcartslidingoninclinedplane ............ 74 4.12Schematictoexplaintheangularangle. ................ 74 4.13Rectangularareaunderpressure. ................... 75 4.14Schematicofsubmergedareatoexplainthecenterforcesandmo-ments ................................... 77 4.15Thegeneralforcesactingonsubmergedarea. ............ 78 4.16Thegeneralforcesactingonnonsymmetricalstraightarea. ..... 79 4.17Thegeneralforcesactingonnonsymmetricalstraightarea. ..... 80 4.18Theeffectsofmultilayersdensityonstaticforces. .......... 83 4.19Theforcesoncurvedarea. ....................... 85 4.20SchematicofNetForceonoatingbody. ................ 86 4.21Damisapartofacircularshape. .................... 87 4.22Areaabovethedamarcsubtracttriangle. ............... 87 4.23Areaabovethedamarccalculationforthecenter. .......... 88 4.24MomentonarcelementaroundPointO. ............... 89 4.25PolynomialshapedamdescriptionforthemomentaroundpointOandforcecalculations. .......................... 90 4.26Thedifferencebetweentheslopandthedirectionangle. ....... 90 4.27SchematicofImmersedCylinder. .................... 92 4.28TheoatingforcesonImmersedCylinder. ............... 93 4.29Schematicofathinwalloatingbody. ................. 94 4.30Schematicofoatingbodies. ...................... 98 4.31Schematicofoatingcubic. ....................... 98 4.32Stabilityanalysisofoatingbody. .................... 99 4.33Cubicbodydimensionsforstabilityanalysis. .............. 100 4.34Stabilityofcubicbodyinnitylong. ................... 101 4.35Themaximumheightreverseasafunctionofdensityratio. ..... 102

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4.36Theeffectsofliquidmovementonthe .............. 103 4.37MeasurementofGMofoatingbody. .................. 105 4.38Calculationsof .............. 106 4.39Aheavyneedleisoatingonaliquid. ................. 107 4.40DescriptionofdepressiontoexplaintheRayleighTaylorinstability. 108 4.41Descriptionofdepressiontoexplaintheinstability. .......... 110 4.42Thecrosssectionoftheinterface.Thepurplecolorrepresentsthemaximumheavyliquidraisingarea.Theyellowcolorrepresentsthemaximumlighterliquidthataregoingdown. ............. 111 5.1linesastaticuidwithaconstantdensity. ............... 115 5.2Stratiedowinhorizontaltubeswhentheliquidsowisveryslow. 117 5.3KindofStratiedowinhorizontaltubes. ............... 118 5.4Plugowinhorizontaltubeswhentheliquidsowisfasterslow. .. 118 5.5ModiedMandhanemapforowregimeinhorizontaltubes. ..... 119 5.6linesastaticuidwithaconstantdensity. ............... 120 5.7Adimensionalverticalowmapunderverylowgravityagainstthegravity. ................................... 121 5.8Theterminalvelocitythatleftthesolidparticles. ............ 131 5.9Theowpatternsinsolid-liquidow. .................. 132 5.10Countercurrentowinacan(theleftgure)hasonlyoneholethuspulseowandaowwithtwoholes(rightpicture). .......... 134 5.11Counterowinverticaltubesmap. ................... 134 5.12PicturesofCounter-currentowinliquidgasandsolidgascong-urations.Thecontainerismadeoftwocompartments.Theuppercompartmentislledwiththeheavyphase(liquid,watersolution,orsmallwoodparticles)byrotatingthecontainer.Eventhoughthesolidgasratioissmaller,itcanbenoticedthatthesolidgasisfasterthantheliquidgasow. .................... 135 5.13Floodinverticalpipe. .......................... 135 5.14Aowmaptoexplainthehorizontalcountercurrentow. ...... 136 5.15Adiagramtoexplaintheoodinatwodimensiongeometry. ..... 137 5.16Generalforcesdiagramtocalculatedtheinatwodimensiongeometry. 142

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........................ xxix 1continue .................................. xxx 1.1Sutherland'sequationcoefcients ................... 13 1.2Viscosityofselectedgases ....................... 14 1.3Viscosityofselectedliquids ....................... 14 1.4Propertiesatthecriticalstage ...................... 15 1.5Bulkmodulusforselectedmaterials .................. 20 1.6Thecontactangleforair/waterwithselectedmaterials. ........ 25 1.7Thesurfacetensionforselectedmaterials. .............. 31 1.7continue .................................. 32 2.1PropertiesofVariousIdealGases[300K] ............... 38 3.1MomentsofInertiaforvariousplanesurfacesabouttheircenterofgravity(fullshapes) ............................ 53 3.2Momentofinertiaforvariousplanesurfacesabouttheircenterofgravity ................................... 54 ix

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1.1 ,seeequa-tion(1.17),page12

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email@provider.netTypocorrectionsandotherminorcontributions

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addingaquestionandperhapsthesolution.Thus,thismethodisexpectedtoacceleratethecreationofthesehighqualitybooks.Thesebooksarewritteninasimilarmannertotheopensourcesoftwareprocess.Someonehastowritetheskeletonandhopefullyotherswilladdeshandskin.Inthisprocess,chaptersorsectionscanbeaddedaftertheskeletonhasbeenwritten.Itisalsohopedthatotherswillcontributetothequestionandanswersectionsinthebook.Butmorethanthat,otherbookscontaindata Progress Remarks Version AvailabilityforPublicDownload beta 0.4.8.2 alpha 0.0.3 NSY 0.0.0 alpha 0.1.8 NSY BasedonEckert 0.0.0 NSY 0.0.0

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Progress Remarks Version AvailabilityforPublicDownload NSY 0.0.0 earlyalpha rstchapter 0.0.1 NSY 0.0.0 earlyalpha 0.0.1 NSY Tel-Aviv'notes 0.0.0

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suchastheLinuxDocumentationprojectdemonstratedthatbookscanbewrittenasthecooperativeeffortofmanyindividuals,manyofwhomvolunteeredtohelp.Writingatextbookiscomprisedofmanyaspects,whichincludetheac-tualwritingofthetext,writingexamples,creatingdiagramsandgures,andwritingtheLATEXmacros

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4 ).Ihopethismakesthebookeasiertouseasareferencemanual.However,thismanuscriptisrstandforemostatextbook,andsecondlyareferencemanualonlyasaluckycoincidence.Ihavetriedtodescribewhythetheoriesarethewaytheyare,ratherthanjustlistingseveneasystepsforeachtask.Thismeansthatalotofinformationispresentedwhichisnotnecessaryforeveryone.Theseexplanationshavebeenmarkedassuchandcanbeskipped.

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1.1 )forthecomplexrelationshipsbetweenthedifferentbrancheswhichonlypartofitshouldbedrawninthesametime.).Forexample,glassap-pearsasasolidmaterial,butacloserlookrevealsthattheglassisaliquidwithalargeviscosity.AproofoftheglassliquidityisthechangeoftheglassthicknessinhighwindowsinEuropeanChurchesafterhundredyears.Thebottompartoftheglassisthickerthanthetoppart.Materialslikesand(somecallitquicksand)andgrainsshouldbetreatedasliquids.Itisknownthatthesematerialshavetheabilitytodrownpeople.Evenmaterialsuchasaluminumjustbelowthemushyzonealsobehavesasaliquidsimilarlytobutter.Afteritwasestablishedthattheboundariesofuidmechanicsaren'tsharp,thediscussioninthisbookislimitedtosimpleand(mostly)Newtonian(sometimespoweruids)uidswhichwillbedenedlater. Theuidmechanicsstudyinvolvemanyeldsthathavenoclearboundarybetweenthem.Researchersdistinguishbetweenorderlyowandchaoticowasthelaminarowandtheturbulentow.Theuidmechanicscanalsobedistinguishbetweenasinglephaseowandmultiphaseow(owmademorethanonephaseorsingledistinguishablematerial).Thelastboundary(asalltheboundariesinuid 1

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liquidassumingacomplexturbulentowmodel.Suchabsurdanalysisarecom-monamongengineerswhodonotknowwhichmodelcanbeapplied.Thus,oneofthemaingoalsofthisbookistoexplainwhatmodelshouldbeapplied.Beforedealingwiththeboundaries,thesimpliedprivatecasesmustbeexplained.Therearetwomainapproachesofpresentinganintroductionofuidme-chanicsmaterials.Therstapproachintroducestheuidkinematicandthenthebasicgoverningequations,tobefollowedbystability,turbulence,boundarylayerandinternalandexternalow.ThesecondapproachdealswiththeIntegralAnal-ysistobefollowedwithDifferentialAnalysis,andcontinuewithEmpiricalAnalysis.Thesetwoapproachesposeadilemmatoanyonewhowritesanintroductorybookfortheuidmechanics.Thesetwoapproacheshavejusticationsandpositivepoints.Reviewingmanybooksonuidmechanicsmadeitclear,thereisn'taclearwinner.Thisbookattemptstondahybridapproachinwhichthekinematicispresentedrst(asidetostandardinitialfourchapters)followbyIntegralanalysisandcontinuedbyDifferentialanalysis.Theidealow(frictionlessow)shouldbeexpandedcomparedtotheregulartreatment.Thisbookisuniqueinprovidingchapteronmultiphaseow.Naturally,chaptersonopenchannelow(asasubclassofthemultiphaseow)andcompressibleow(withthelatestdevelopments)areprovided.

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thecomputersduringthe60sandmuchmorepowerfulpersonalcomputerhaschangedtheeld.Therearemanyopensourceprogramsthatcananalyzemanyuidmechanicssituations.Todaymanyproblemscanbeanalyzedbyusingthenumericaltoolsandprovidereasonableresults.Theseprogramsinmanycasescancapturealltheappropriateparametersandadequatelyprovideareasonabledescriptionofthephysics.However,therearemanyothercasesthatnumericalanalysiscannotprovideanymeaningfulresult(trends).Forexample,noweatherpredictionprogramcanproducegoodengineeringqualityresults(wherethesnowwillfallwithin50kilometersaccuracy.Buildingacarwiththisaccuracyisadis-aster).Inthebestscenario,theseprogramsareasgoodastheinputprovided.Thus,assumingturbulentowforstillowsimplyprovideserroneousresults(seeforexample,EKK,Inc). 1.1 ).Thestudyofthiskindofmaterialcalledrheologyanditwill(almost)notbediscussedinthisbook.Itisevidentfromthisdiscussionthatwhenaliquidisatrest,noshearstressisapplied.Theuidismainlydividedintotwocategories:liquidsandgases.Themaindifferencebetweentheliquidsandgasesstateisthatgaswilloccupythewholevolumewhileliquidshasanalmostxvolume.Thisdifferencecanbe,formostpracticalpurposesconsidered,sharpeventhoughinrealitythisdifferenceisn'tsharp.Thedifferencebetweenagasphasetoaliquidphaseabovethecrit-icalpointarepracticallyminor.Butbelowthecriticalpoint,thechangeofwaterpressureby1000%onlychangethevolumebylessthan1percent.Forexample,achangeinthevolumebymore5%willrequiredtensofthousandspercentchangeofthepressure.So,ifthechangeofpressureissignicantlylessthanthat,thenthechangeofvolumeisatbest5%.Hence,thepressurewillnotaffectthevolume.Ingaseousphase,anychangeinpressuredirectlyaffectsthevolume.Thegasllsthevolumeandliquidcannot.Gashasnofreeinterface/surface(sinceitdoeslltheentirevolume).Thereareseveralquantitiesthathavetobeaddressedinthisdiscussion.Therstisforcewhichwasreviewedinphysics.Theunitusedtomeasureis[N].Itmustberememberthatforceisavector,e.githasadirection.Thesecondquantitydiscussedhereisthearea.Thisquantitywasdiscussedinphysicsclassbuthereithasanaddtionalmeaning,anditisreferredtothedirectionofthearea.Thedirectionofareaisperpendiculartothearea.Theareaismeasuredin[m2].Area

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1.2 showsthedensityasafunctionofthesam-plesize.Aftercertainsamplesize,thedensityremainsconstant.Thus,thedensityisdenedas=limV!"m 1.2 forpointwherethegreenlinescon-vergetoconstantdensity).Whenthisassumptionisbroken,then,theprinciplesofstatisticalmechanicsmustbeutilized. 1.3 ).TheupperplatevelocitygenerallywillbeU=f(A;F;h)

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WhereAisthearea,theFdenotestheforce,histhedistancebetweentheplates.Fromsolidmechanicsstudy,itwasshownthatwhentheforceperareaincreases,thevelocityoftheplateincreasesalso.Experimentsshowthattheincreaseofheightwillincreasethevelocityuptoacertainrange.Considermovingtheplatewithazerolubricant(h0)(resultsinlargeforce)oralargeamountoflubricant(smallerforce).Inthisdiscussion,theaimistodevelopdifferentialequation,thusthesmalldistanceanalysisisapplicable.Forcaseswherethedependencyislinear,thefollowingcanbewrittenU/hF A 1.3 )canberearrangedtobeU h/F A A 1.4 )and( 1.5 )itfollowsthatratioofthevelocitytoheightispro-portionaltoshearstress.Hence,applyingthecoefcienttoobtainanewequalityasxy=U h 1.4 )itcanbenoticedthatforasmallangle,theregularapproxi-mationprovidesd`=Ut=geometryz }| {h 1.8 )itfollowsthatU=h t

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1.9 )withequation( 1.6 )yieldsxy= t t=dU dy 1.10 )referredtoasNewtonianuid.Forthiskindofsubstancexy=dU dy 1.9 )canbeinterpretedasmomentuminthexdirectiontrans-feredintotheydirection.Thus,theviscosityistheresistancetotheow(ux)orthemovement.Thepropertyofviscosity,whichisexhibitedbyalluids,isduetotheexistenceofcohesionandinteractionbetweenuidmolecules.Thesecohe-sionandinteractionshampertheuxinydirection.Somereferredtoshearstressasviscousuxofxmomentumintheydirection.TheunitsofshearstressarethesameasuxpertimeasfollowingF Akgm sec21 Akg secm sec1 Example1.1: 1.6 ))F=AU h11:0690:5 0:01=53:45[N]

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Example1.2: }| {12=60=0:4riWhererpsisrevolutionpersecond.Thesamewayasinexample( 1.1 ),themomentcanbecalculatedastheforcetimesthedistanceasM=F`=riz}|{`2rihz}|{AU roriInthiscaserori=hthus,M=22riz}|{0:13hz }| {0:9860:4

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1.6 demonstratesthatviscosityincreasesslightlywithpressure,butthisvariationisnegligibleformostengineeringproblems.Wellabovethecriticalpoint,bothmaterialsareonlyafunctionofthetemperature.Ontheliquidsidebelowthecriticalpoint,thepressurehasminoreffectontheviscosity.Itmustbestressthattheviscosityinthedomeismeaningless.Thereisnosuchathingofviscosityat30%liquid.Itsimplydependsonthestructureoftheowaswillbediscussedinthechapteronmultiphaseow.Thelinesintheabovediagramsareonlytoshowconstantpressurelines.Oilshavethegreatestincreaseofviscositywithpressurewhichisagoodthingformanyengineeringpurposes. 1.5 ),therelationshipbetweenthevelocityandtheshearstresswasassumedtobelinear.Notallthematerialsobeythisrelationship.Thereisalargeclassofmaterialswhichshowsanon-linearrelationshipwithvelocityforanyshearstress.Thisclassofmaterialscanbeapproximatedbyasinglepolynomialtermthatisa=bxn.Fromthephysicalpointofview,thecoefcientdependsonthevelocitygradient.Thisrelationshipisreferredtoaspowerrelationshipanditcan

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bewrittenas=viscosityz }| {KdU dxn1dU dx 1.13 )areconstant.Whenn=1equationrepresentNewtonianuidandKbecomesthefamiliar.Theviscositycoefcientisalwayspositive.Whenn,isaboveone,theliquidisdilettante.Whennisbe-lowone,theuidispseudoplastic.Theliquidswhichsatisfyequation( 1.13 )arereferredtoaspurelyviscousuids.Manyuidssatisfytheaboveequation.Flu-idsthatshowincreaseintheviscosity(withincreaseoftheshear)referredtoasthixotropicandthosethatshowdecreasearecalledreopecticuids(seeFigure 1.5 ). 1.7 .However,formostpracticalpurposes,thiskindofguresisn'tusedinregularengineeringpractice.

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1.8 and 1.9 T03 2 WhereviscosityatinputtemperatureT0referenceviscosityatreferencetemperature,Ti0TininputtemperatureindegreesKelvinTi0referencetemperatureindegreesKelvinSuthSuthisSutherland'sconstantanditispresentedintheTable 1.1 Example1.3: 1.1 .SOLUTION 0:555800+120800 524:073 22:51105Nsec m2

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chemicalformula Sutherland 527.67 0.00000982 standardair 120 524.07 0.00001827 carbondioxide 527.67 0.00001480 carbonmonoxide 518.67 0.00001720 hydrogen 528.93 0.0000876 nitrogen 540.99 0.0001781 oxygen 526.05 0.0002018 sulfurdioxide 528.57 0.0001254 1.10 exhibitsseveralliquidmetals(fromTheRe-actorHandbook,Vol.AtomicEnergyCommissionAECD-3646U.S.GovernmentPrintingOfce,WashingtonD.C.May1995p.258.)TheGeneralViscosityGraphsIncaseordinaryuidswhereinformationislimit,Hougenetalsuggestedtousegraphsimilartocompressibilitychart.Inthisgraph,ifonepointiswelldocumented,otherpointscanbeestimated.Furthermore,thisgraphalsoshowsthetrends.InFigure 1.11 therelativeviscosityr==cisplottedasafunction

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formula TemperatureT[C] m2] 0.0000076 0.0000109 0.0000146 oxygen 0.0000203 mercuryvapor 0.0000654 formula TemperatureT[C] m2] 0.000245 0.000647 0.000946 0.001194 0.001547 0.01915 OliveOil 25 0.084 CastorOil 25 0.986 Clucuse 25 5-20 CornOil 20 0.072 SAE30 0.15-0.200 SAE50 SAE70 Ketchup Ketchup Benzene Firmglass 20 1.069

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chemicalcomponent MolecularWeight m2] 33.3 12.9696 3.47 5.26 2.289945 2.54 44.5 27.256425 15.6 151 48.636 26.4 289.8 58.7685 49. Airmix 28.97 132 36.8823 19.3 304.2 73.865925 19.0 154.4 50.358525 18.0 305.4 48.83865 21.0 190.7 46.40685 15.9 Water 647.096K 22.064[MPa] 1.4 orsimilarinformation.Thesecondway,iftheinformationisavailableandiscloseenoughtothecriticalpoint,thenthecriticalviscosityisobtainedasc=givenz}|{ r|{z}gure 1.11 1.11 obtainthereducedviscosity. Example1.4:

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m2ThevalueofthereducedtemperatureisTr373:15 154:42:41ThevalueofthereducedpressureisPr20 50:350:4FromFigure 1.11 itcanbeobtainedr1:2andthepredictedviscosityis=cTablez }| { c=181:2=21:6[Nsec=m2]Theobservedvalueis24[Nsec/m2] Example1.5:

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i Component MolecularWeight,M 0.2 0.0000203 2 0.8 0.00001754

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j 1 1.0 1.0 1.0 2 1.143 1.157 1.0024 2 1 0.875 .86 0.996 2 1.0 1.0 1.0 0:21:0+0:81:0024+0:80:00001754 0:20:996+0:81:00:0000181Nsec m2Theobservedvalueis0:0000182Nsec m2.Inverylowpressure,intheory,theviscosityisonlyafunctionofthetem-

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peraturewithasimplemolecularstructure.Forgaseswithverylongmolecularstructureorcomplexitystructuretheseformulascannotbeapplied.Forsomemix-turesoftwoliquidsitwasobservedthatatalowshearstress,theviscosityisdominatedbyaliquidwithhighviscosityandathighshearstresstobedominatedbyaliquidwiththelowviscosityliquid.Thehigherviscosityismoredominateatlowshearstress.ReinerandPhillippoffsuggestedthefollowingformuladUx m2,0=0:00105Nsec m2,ands=0:0000073kN m2.Thisequation( 1.23 )providesreasonablevalueonlyupto=0:001kN m2.Figure 1.12 canbeusedforacrudeestimateofdensegasesmixture.Toes-timatetheviscosityofthemixturewithncomponentHougenandWatson'smethodforpseudocritialpropertiesisadapted.ItthismethodthefollowingisdenedasPcmix=nXi=1xiPci @vT

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1.27 )intoBT=@P @T 1.5 BulkModulus109N m 2.49 593K 57.8[Bar] Acetone 0.80 508K 48[Bar] Benzene 1.10 562K 4.74[MPa] CarbonTetrachloride 1.32 556.4K 4.49[MPa] EthylAlcohol 1.06 514K 6.3[Mpa] Gasoline 1.3 nf nf Glycerol 4.03-4.52 850K 7.5[Bar] Mercury 26.2-28.5 1750K 172.00[MPa] MethylAlcohol 0.97 Est513 Est78.5[Bar] Nitrobenzene 2.20 nf nf OliveOil 1.60 nf nf ParafnOil 1.62 nf nf SAE30Oil 1.5 na na Seawater 2.34 na na Toluene 1.09 591.79K 4.109[MPa] Turpentine 1.28 na na Water 2.15-2.174 647.096K 22.064[MPa] Intheliterature,additionalexpansionsforsimilarparametersaredened.ThethermalexpansionisdenedasP=1 @TP @Tv

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Thisparameterindicatesthechangeofthepressureduetothechangeoftem-perature(wherev=constant).Thesedenitionsarerelatedtoeachother.ThisrelationshipisobtainedbytheobservationthatthepressureasafunctionofthetemperatureandspecicvolumeasP=f(T;v) @TvdT+@P @vTdv 1.32 )is0=@P @TvdT+@P @vTdv 1.33 )followsthatdv dTP=const=@P @Tv @vT 1.34 )indicatesthatrelationshipforthesethreecoefcientsisT=v 1.35 )sometimesisusedinmeasurementofthebulkmodulus. Example1.6:

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@vv 0:0003514285:714[Bar] v2:15109:01=2:15107[N=m2]=215[Bar]

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Foraverysmallarea,theanglesareverysmallandthus(sin).Fur-thermore,itcanbenoticedthatd`i2Ridi.Thus,theequation( 1.36 )canbesimpliedasP=1 1.37 )predictsthatpressuredifferenceincreasewithinverseofthera-dius.Therearetwoextremecases:one)radiusofinniteandradiusofnitesize.Thesecondwithtwoequalradii.Therstcaseisforaninnitelongcylinderforwhichtheequation( 1.37 )isreducedtoP=1 1.37 )isreducedtoP=2 R R Example1.8: 1.39 )forreversibleprocess.Hencetheworkisw=Zrfr0Pz}|{2 rdvz }| {4r2dr=8Zrfr0rdr=4rf2r02

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1.14 ,forcesdiagramisshownwhencontrolvolumeischosensothatthemassesofthesolid,liq-uid,andgascanbeignored.Regardlesstothemagnitudeofthesurfacetensions(excepttozero)theforcescannotbebalancedforthedescriptionofstraightlines.Forexample,forcesbalancedalongthelineofsolidboundaryisgslslgcos=0 1.44 )intoequation( 1.43 )yieldsgsls=Fsolid For==2=)tan=1.Thus,thesolidreactionforcemustbezero.Thegassolidsurfacetensionisdifferentfromtheliquidsolidsurfacetensionandhenceviolatingequation( 1.43 ).

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Theconnectionofthethreephasesmaterialsmediumscreatestwositua-tionswhicharecategorizedaswettingornonwetting.Thereisacommonde-nitionofwettingthesurface.Iftheangleofthecontactbetweenthreematerialsislargerthan90thenitisnon-wetting.Ontheotherhand,iftheangleisbelowthan90thematerialiswettingthesurface(seeFigure 1.15 ).Theangleisdeter-minedbypropertiesoftheliquid,gasmediumandthesolidsurface.Andasmallchangeonthesolidsurfacecanchangethewettingconditiontononwetting.Infacttherearecommercialspraysthatareintenttochangethesurfacefromwet-tingtononwetting.Thisfactisthereasonthatnoreliabledatacanbeprovidedwiththeexceptiontopuresubstancesandperfectgeometries.Forexample,waterisdescribedinmanybooksasawettinguid.Thisstatementiscorrectinmostcases,however,whensolidsurfaceismadeorcottedwithcertainmaterials,thewaterischangedtobewetting(forexample3Msellingproducttochangewatertononwetting).So,thewetnessofuidsisafunctionofthesolidaswell. ContactAngle Source Steel Steel,Nickel Nickel Nickel Chrome-NickelSteel Silver Zink Bronze Copper Copper Copper 1 R.Siegel,E.G.Keshock(1975)Effectsofreducedgravityonnucleateboilingbubbledynamicsinsaturatedwater,AIChEJournalVolume10Issue4,Pages509-517.1975 2 BerglesA.E.andRohsenowW.M.Thedeterminationofforcedconvectionsurfaceboilingheattransfer,ASME,J.HeatTransfer,vol1pp365-372. 3 Tolubinsky,V.I.andOstrovsky,Y.N.(1966)Onthemechanismofboilingheattransfer,.InternationalJournalofHeatandMassTransfer,Vol.9,No12,pages1465-1470. 4 ArefevaE.I.,AladevO,I.T.,(1958)wlijaniismatchivaemostinateploobmenpri

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5 LabuntsovD.A.(1939)Approximatetheoryofheattransferbydevelopednu-cleateboilingInSussianIzvestiyaAnSSSR,EnergetikaItransport,No1. 6 Basu,N.,Warrier,G.R.,andDhir,V.K.,(2002)OnsetofNucleateBoilingandActiveNucleationSiteDensityduringSubcooledFlowBoiling,ASMEJournalofHeatTransfer,Vol.124,papes717-728. 7 Gaetner,R.F.,andWestwater,J.W.,(1960)PopulationofActiveSitesinNu-cleateBoilingHeatTransfer,Chem.Eng.Prog.Symp.,Ser.56. 8 Wang,C.H.,andDhir,V.K.,(1993),EffectofSurfaceWettabilityonActiveNucleationSiteDensityDuringPoolBoilingofWateronaVerticalSurface,J.HeatTransfer115,pp.659-669Toexplainthecontourofthesurface,andthecontactangleconsidersimplewettingliquidcontactingasolidmaterialintwodimensionalshapeasdepictedinFigure 1.16 .Tosolvetheshapeoftheliquidsurface,thepressuredifferencebetweenthetwosidesoffreesurfacehastobebalancedbythesurfacetension.InFigure 1.16 describestheraisingoftheliquidasresultsofthesurfacetension.Thesurfacetensionreducesthepressureintheliquidabovetheliquidline(thedottedlineintheFigure 1.16 ).Thepressurejustbelowthesurfaceisgh(x)(thispressuredifferencewillbeexplainedinmoredetailsinChapter 4 ).Thepressure,onthegasside,istheatmosphericpressure.Thisproblemisatwodimensionalproblemandequation( 1.38 )isapplicabletoit.Appallingequation( 1.38 )andusingthepressuredifferenceyieldsgh(x);= R(x) 1.47 )canbederivedeitherbyforcingacircleatthreepointsat(x,x+dx,andx+2dx)andthusndingthethediam-eterorbygeometricalanalysisoftriangles

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buildonpointsxandx+dx(perpendiculartothetangentatthesepoints).Substi-tutingequation( 1.47 )intoequation( 1.46 )yieldsgh(x)= 1.50 )isnonlineardifferentialequationforheightandcanbewrittenas 1+dh dx2!3=2d2h dx2=0 1.50 )isgh=h 1.50 )transformsintoZg hdh=Zh 1.51 )intoZ1 1.52 )becomesh2

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1.54 )isarstorderdifferentialequationthatcanbesolvedbyvariablesseparation 1.54 )canberearrangedtobe1+_h21=2=1 1h2 1h2 dx=vuut 1 1h2 1 1h2 1.59 )canbeintegratedtoyield8->>>>>>>>>>>>>>>>>>>;-dh 1 1h2

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gr 1.60 )when=0andthuscos=1.Thisangleisobtainedwhenaperfecthalfasphereshapeexistoftheliquidsurface.Inthatcaseequation( 1.60 )becomeshmax=2 gr 1.18 exhibitstheheightasafunctionoftheradiusofthetube.Theheightbasedonequation( 1.61 )isshowninFigure 1.17 asblueline.Theactualheightisshownintheredline.Equation( 1.61 )providesreasonableresultsonlyinacertainrange.Forasmalltuberadius,equation( 1.49 )provedbetterresultsbe-causethecurveapproacheshemisphericalshaper(smallgravityeffect).Forlargeradiiequation( 1.49 )approachesthestraitline(theliquidline)stronggravityeffect.Ontheotherhand,forextremelysmallradiiequation( 1.61 )indicatesthatthehighheightwhichindicatesanegativepressure.Theliquidatacertainpressurewillbevaporizedandwillbreakdownthemodeluponthisequationwasconstructed.Fur-thermore,thesmallscaleindicatesthatthesimplisticandcontinuousapproachisnotappropriateandadifferentmodelisneeded.TheconclusionofthisdiscussionareshowninFigure 1.17 .Theacutaldimensionformanyliquids(evenwater)isabout1-5[mm].Thediscussionabovewasreferredtowettingcontactangle.Thedepres-sionoftheliquidoccursinanegativecontactanglesimilarlytowetting.Thedepressionheight,hissimilartoequation( 1.61 )withaminussign.However,the 1.6

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1.61 ).ThemeasurementsoftheheightofdistilledwaterandmercuryarepresentedinFigure 1.18 .Theexperimentalresultsofthesematerialsarewithagreementwiththediscussionabove.ThesurfacetensionofaselectedmaterialisgiveninTable 1.7 .Inconclusion,thesurfacetensionissueisimportantonlyincasewheretheradiusisverysmallandgravityisnegligible.Thesurfacetensiondependsonthetwomaterialsormediumsthatitseparates. Example1.9: 1.39 ).D=2R=22 10002:912104[m] r20:0728 0:0002728:0[N=m2]

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SurfaceTensionmN m mK 27.6 Acetone 25.20 -0.1120 Aniline 43.4 Benzene 28.88 -0.1291 Benzylalcohol 39.00 -0.0920 Benzylbenzoate 45.95 -0.1066 Bromobenzene 36.50 -0.1160 Bromobenzene 36.50 -0.1160 Bromoform 41.50 -0.1308 Butyronitrile 28.10 -0.1037 Carbondisuld 32.30 -0.1484 Quinoline 43.12 -0.1063 Chlorobenzene 33.60 -0.1191 Chloroform 27.50 -0.1295 Cyclohexane 24.95 -0.1211 Cyclohexanol 34.40 Cyclopentanol 32.70 -0.1011 CarbonTetrachloride 26.8 n/a Carbondisuld 32.30 -0.1484 Chlorobutane 23.10 -0.1117 EthylAlcohol 22.3 n/a Ethanol 22.10 -0.0832 Ethylbenzene 29.20 -0.1094 Ethylbromide 24.20 -0.1159 Ethyleneglycol 47.70 -0.0890 Formamide 58.20 -0.0842 Gasoline n/a Glycerol 64.0 -0.0598 Helium 0.12 Mercury 425-465.0 -0.2049 Methanol 22.70 -0.0773 Methylnaphthalene 38.60 -0.1118 MethylAlcohol 22.6 n/a Neon 5.15 Nitrobenzene 43.90 -0.1177 OliveOil 43.0-48.0 -0.067 Peruoroheptane 12.85 -0.0972 Peruorohexane 11.91 -0.0935

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SurfaceTensionmN m mK 14.00 -0.0902 Phenylisothiocyanate 41.50 -0.1172 Propanol 23.70 Pyridine 38.00 -0.1372 Pyrrol 36.60 -0.1100 SAE30Oil n/a n/a Seawater 54-69 n/a Toluene 28.4 -0.1189 Turpentine 27 n/a Water 72.80 -0.1514 o-Xylene 30.10 -0.1101 m-Xylene 28.90 -0.1104

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33

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2.3 )isthatthewaytheworkisdoneand/orinterme-diatestatesareirrelevanttonalresults.Thereareseveraldenitions/separationsofthekindofworksandtheyincludekineticenergy,potentialenergy(gravity),chemicalpotential,andelectricalenergy,etc.Theinternalenergyistheenergythatdependsontheotherpropertiesofthesystem.Forexampleforpure/homogeneousandsimplegasesitdependsontwopropertiesliketemperatureandpressure.TheinternalenergyisdenotedinthisbookasEUanditwillbetreatedasastateprop-erty.Thepotentialenergyofthesystemisdependedonthebodyforce.Acommonbodyforceisthegravity.Forsuchbodyforce,thepotentialenergyismgzwheregisthegravityforce(acceleration),misthemassandthezistheverticalheightfromadatum.ThekineticenergyisK:E:=mU2 2.5 )istransformedinto

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Sincetheaboveequationsaretruebetweenarbitrarypoints,choosinganypointintimewillmakeitcorrect.Thusdifferentiatingtheenergyequationwithrespecttotimeyieldstherateofchangeenergyequation.TherateofchangeoftheenergytransferisDQ Dt=_Q Dt=_W Dt+mDBfz Dt 2.9 )reducedto Dt+mgDz Dt T0 T=0 Trev (2.13)

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Trev=Z21dS 2.12 )canbewrittenasQ=TdS 2.15 )( 2.16 )into( 2.10 )resultsinTdS=dEU+PdV 2.18 )the( 2.17 )yields

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Forisentropicprocess,equation( 2.17 )isreducedtodH=VdP.Theequation( 2.17 )inmassunitisTds=du+Pdv=dhdP @T kmolK

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M 2.1 KgKiCvhkj KgKiCPhkj KgKik ArgonAr39.9480.208130.52030.31221.400 ButaneC4H1058.1240.143041.71641.57341.091 CarbonDioxideCO244.010.188920.84180.65291.289 CarbonMonoxideCO28.010.296831.04130.74451.400 EthaneC2H630.070.276501.76621.48971.186 EthyleneC2H428.0540.296371.54821.25181.237 HeliumHe4.0032.077035.19263.11561.667 HydrogenH22.0164.1241814.209110.08491.409 MethaneCH416.040.518352.25371.73541.299 NeonNe20.1830.411951.02990.61791.667 NitrogenN228.0130.296801.04160.74481.400 OctaneC8H18114.2300.072791.71131.63851.044 OxygenO231.9990.259830.92160.66181.393 PropaneC3H844.0970.188551.67941.49091.327 SteamH2O18.0150.481521.87231.41081.327 Fromequation( 2.25 )ofstateforperfectgasitfollowsd(Pv)=RdT (2.29)

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Fromthedenitionofenthalpyitfollowsthatd(Pv)=dhdEu 2.28 )andsubsistingintoequation( 2.30 )anddividingbydTyieldsCpCv=R k1 k1 2.1 ).Theentropyforidealgascanbesimpliedasthefollowings2s1=Z21dh TdP T TZ21RdP P=CplnT2 2.35 )transformedintos2s1 k1lnT2 2.37 )as

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RT

PAGE 83

41

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}| {(x)dV 3.1 .Also,themass,misthetotalmassoftheobject.Itcanbenoticedthatcenterofmassinthexdirectionisn'taffectedbythedistributionintheynorbyzdi-rections.Insamefashionthecenterofmasscanbedenedintheotherdirectionsasfollowing 3.2 whichhasdensity,.Thus,equation( 3.1 )canbetransferedintox=1 }| {tdA 3.3 )canbetransferedinto

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whentheintegralnowoveronlytheareaasopposeoverthevolume.Findingthecenteroidlocationshouldbedoneinthemostconvenientcoordi-natesystemsincethelocationiscoordinateindependent. 3.5 )canbetransformedintoIrrm=ZVr2dV

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}| {ZAr2dA 3.10 )canbewrittenasIxx=ZAy2+z2dA

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equation( 3.15 )canbeexpendedasIx0x0=Ixxz }| {ZAy2+z2dA+=0z }| {2ZA(yy+zz)dA+ZA(y)2+(z)2dA 3.16 )ontherighthandsideisthemomentofinertiaaboutaxisxandthesecondthemiszero.Thesecondthermiszerobecauseitintegralofcenteraboutcenterthusiszero.ThethirdtermisanewtermandcanbewrittenasZAconstantz }| {(y)2+(z)2dA=r2z }| {(y)2+(z)Az }| {Z2AdA=r2A 3.4 andtherefore,Ixx=nXi=1Ixxi 3.20 )isveryusefulinthecalculationofthemomentofinertiautilizingthemomentofinertiaofknownbodies.Forexample,themomentofin-ertialofhalfacircleishalfofwholecir-cleforaxisathecenterofcircle.Themomentofinertiacanthenmovethecenterofarea.ofthe

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3.5 .Thematerialiswithanuniformdensityandhomogeneous.SOLUTION 3.5 asIrr=ZVr2dm=Zr00r2dVz }| {h2rdr=h2r04 2hr04=1 2mr02Theradiusofgyrationisrk=s 2mr02 3.6 .SOLUTION 3.12 )asfollowingIxx=ZA0B@0z}|{y2+z21CAdA=Za0z2dAz}|{bdz=a3b

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isdIxxm=tz}|{dyIxxz }| {ba3 }| {ba3 }| {z2|{z}r2ba|{z}A1CCCA 3.22 )towriteasIxxm=Zt=2t=2ba3 Ixxm=ba3 1+t2 3.24 )indicatesthatratioap-proachesonewhenthicknessratioisapproacheszero,Ixxm(t!0)!1.Ad-ditionallyitcanbenoticedthattheratioa2=t2istheonlycontributortotheer-ror 3.8 .Icanbenoticedthattheerrorissignicantveryfastevenforsmallval-uesoft=awhilethewithofthebox,bhasnoeffectontheerror. Example3.4: 3.9

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}| {(b2)d=2(31) 3b 3 2Thecenterofareacanbecalculateduti-lizingequation( 3.4 ).Thecenterofev-eryelementisat,2+b2 }| {2+(b2) 2dAz }| {(b2)d=3b 3.25 )canbedoneintwostepsrstcalculatethemomentofinertiainthiscoor-dinatesystemandthenmovethecoordinatesystemtocenter.Utilizingequation( 3.12 )anddoingtheintegrationfrom0tomaximumyprovidesIx0x0=4Zb02dAz }| {r d=2b7=2 3.18 )Ixx=Ix0x0Ax2=Ix0x0z }| {4b7=2 }| {31 3b 3 2(x=xc)2z }| {3b

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Forexample,theproductofinertiaforxandyaxisesisIxy=ZAxydA 3.28 )resultsinIx0y0=Ixyz }| {ZAxydA+y0z }| {ZAxdAz }| {ZAxydA+x0z }| {ZAydAz }| {ZAxydA+xyAz }| {ZAxydA Example3.5: bx+aTheproductofinertiaatthecenteriszero.ThetotalproductofinertiaisIx0y0=0+xz}|{a }| {ab

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3.32 )canbetransformintoIx0x0000Iy0y0000Iz0z0 3.33 )referredasprinciplesystem. 3.34 )canbetransformedtoacontinuousformasXF=ZVD(U) DtZVUdV

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Thevelocity,Uisaderivativeofthelocationwithrespecttotime,thus,XF=D2 Dt=D Dt(rUdm) Dt=D DtZm(rUdm) Dt(rU)=D Dt(rDr Dt)=D2r 3.38 )providesL=rU=0@^i^j^kxy0uv01A=(xvyu)^k

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3.40 )tocalculatethetorqueasT=rF=0@^i^j^kxy0FxFy01A=(xFxyFy)^k Dt[(xvyu)dm] DtZmrUdm

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Picturedescription ab3 3b 3 2

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Picturedescription r2 9) ab 9) ab 9) 2r2 4(1 2sin2) 3rsin 4(+1 2sin2)

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4.1 .Thesystemisinabodyforceeld,gG(x;y;z).ThecombinationofanaccelerationandthebodyforceresultsineffectivebodyforcewhichisgGa=geff 4.1 )canbereducedandsimpliedforthecaseofnoacceleration,a=0.Inthesederivations,severalassumptionsmustbemade.Therstassump-tionisthatthechangeinthepressureisacontinuousfunction.Thereisnorequire-mentthatthepressurehastobeamonotonousfunctione.g.thatpressurecan 55

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@xdydx^i @x^i+@P @y^j+@P @y^k 4.3 )referredtointheliteratureasthepressuregradient.Thismathematicaloperationhasageometricalinterpreta-tion.Ifthepressure,P,wasatwodimensionalheight(thatisonlyafunctionofxandy)thenthegradientisthesteepestascentoftheheight(tothevalley).Thesecondpointisthatthegradientisavector(thatis,ithasadirection).Eventhough,thepressureistreated,now,asascalarfunction(therenoreferencetotheshearstressinpartofthepressure)thegradientisavector.Forexample,thedotproductofthefollowingis^igradP=@P @x @n

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or 4.8 )astheFluidStaticEquation.Thisequationcanbeintegratedandthereforesolved.However,thereareseveralphysicalimplica-tionstothisequationwhichshouldbediscussedandarepresentedhere.First,adiscussiononasimpleconditionandwillcontinueinmorechallengingsituations. 4.9 )andsubstitutingitintoequation( 4.8 )resultsintothreesimplepartialdifferentialequations.Theseequationsare@P @x=@P @y=0 @z=g 4.10 )canbeintegratedtoyieldP(x;y)=constant 4.12 )canbeabsorbedbytheintegrationofequation( 4.11 )andthereforeP(x;y;z)=gz+constant Theintegrationconstantisdeterminedfromtheinitialconditionsoranother

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4.13 )be-comesP(z)P0=g(zz0) 4.13 )thatthepressuredependsonlyonzand/ortheconstantpressurelinesareintheplaneofxandy.Figure 4.2 describestheconstantpressurelinesinthecon-tainerunderthegravitybodyforce.Thepressurelinesarecontinuouseveninareawherethereisadiscontinuousuid.Thereasonthatasolidboundarydoesn'tbreakthecontinuityofthepres-surelinesisbecausethereisalwaysapathtosomeoftheplanes.Itisconvenienttoreversethedi-rectionofztogetridofthenegativesignandtodenehasthedependentoftheuidthatish(zz0)soequation( 4.14 )becomes 4.15 )isdenedaspiezo-metricpressure.

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4.3 .Theliquidisllingthetubeandisbroughtintoasteadystate.Thepressureabovetheliquidontherightsideisthevaporpressure.Usingliquidwithaverylowvaporpressurelikemercury,willresultinadevicethatcanmeasurethepressurewithoutadditionalinformation(thetemperature). Example4.1: 4.3 .Themercurydensityis13545.85[kg=m3].SOLUTION 4.15 )canbeutilizedanditcanbenoticedthatpressureatpointaisPa=gh+Pvapor

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4.4 ).OnetechniqueistoattachedUtubetothechamberandmeasurethepressure.Thisway,thegasispreventedfromescapinganditspressurecanbemeasuredwithaminimalinter-ferencetothegas(somegasenterstothetube).Thegasdensityissignicantlylowerthantheliquiddensityandthereforecanbeneglected.ThepressureatpointisP1=Patmos+gh 4.5 showsatypicalandsimpleschematicofsuchaninstrument.IfthepressuredifferencesbetweenP1andP2issmallthisinstrumentcanmagniedheight,h1andprovidebetteraccuracyreading.Thisdeviceisbasedonthefol-lowingmathematicalexplanation.Insteadystate,thepressurebalance(onlydifferences)isP1+g1(h1+h2)=P2+gh22 4.18 )thegasdensitywasneglected.ThepressuredifferencecanbeexpressedasP1P2=g[2h21(h1+h2)]

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Ifthelightliquidvolumeinthetwocontainersisknown,itprovidestherelationshipbetweenh1andh2.Forexample,ifthevolumesintwocontainersisequalthenh1A1=h2A2!h1=h2A2 4.20 ).Butthisratioeasilycanbeinsertedintothederivations.Withtheequationforheight( 4.20 )equation( 4.18 )becomesP1P2=gh2211A2 4.22 )becomesh2=P1P2 4.23 )asmagnicationfactorsinceitreplacetheregulardensity,2. IdealGasunderHydrostaticPressureThegasdensityvarygraduallywiththepressure.Asrstapproximation,theidealgasmodelcanbeemployedtodescribethedensity.Thusequation( 4.11 )becomes@P @z=gP RT

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P=gdz RT 4.25 )canbeintegratedfrompointtoanypointtoyieldlnP P0=g RT(zz0) 4.26 )tothefollowingP P0=eg(zzo) 4.27 )canbeexpandedtoshowthedifferencetostandardassumptionofconstantpressureasP P0=1h0g P0z }| {(zz0)g RT+(zz0)2g P0=1+0g P00B@hcorrectionfactorz }| {h2 4.29 )isusefulinmathematicalderivationsbutshouldbeignoredforpracticaluse 4.27 )and( 4.28 ).Thecompressibilityisdenedinequation( 2.39 ).ThemodiedequationisP P0=eg(zzo)

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OrinaseriesformwhichisP P0=1(zz0)g ZRT+(zz0)2g 1.28 ).Thesimplestapproachistoassumethatthebulkmodulusisconstant(orhassomerepresentativeaverage).Forthesecases,therearetwodifferentialequationsthatneededtobesolved.Fortunately,here,onlyonehydrostaticequationdependsondensityequation.So,thedifferentialequationfordensityshouldbesolvedrst.Thegoverningdifferentialdensityequation(seeequation( 1.28 ))is=BT@ @P 4.32 )shouldbeseparatedandthentheintegrationcanbecarriedoutasZPP0dP=Z0BTd 4.33 )yieldsPP0=BTln 0 4.34 )canberepresentedinamoreconvenientformas 4.35 )isthecounterpartfortheequationofstateofidealgasfortheliquidphase.Utilizingequation( 4.35 )inequation( 4.11 )transformedinto@P @z=g0ePP0 4.36 )canbeintegratedtoyieldBT

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4.37 )hasunitsoflength.Theintegrationconstant,withunitsoflength,canbeevaluatedatanyspecicpoint.Ifatz=0thepressureisP0andthedensityis0thentheconstantisConstant=BT 4.39 )andisplottedinFigure 4.6 .Thesolutionisareversefunction(thatisnotP=f(z)butz=f(P))itisamonotonousfunctionwhichiseasytosolveforanynumericalvalue(thatisonlyonezcorrespondstoanyPres-sure).Sometimes,thesolutionispre-sentedasP P0=BT BT+1+1 4.40 )ispresentedforhistoricalrea-sonsandinordertocomparetheconstantdensityassumption.Theexponentcanbeexpandedas0BBBBBB@piezometricpressurez }| {(PP0)+correctionsz }| {BT

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Itcanbenoticedthatequation( 4.42 )isreducedtothestandardequationwhenthenormalizedpressureratio,P=BTissmall(<<1).Additionally,itcanbeobservedthatthecorrectionisonthelefthandsideandnotasthetraditionalcorrectiononthepiezometricpressureside. dh=Cx 4.45 )with( 4.11 )resultsin@P @h=gP R(T0Cxh) 4.46 )andchangingtheformal@totheinfor-maldtoobtaindP P=gdh R(T0Cxh)

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P=g RCxd 4.47 )andreusing(thereversedenitions)thevari-ablestransformedtheresultintolnP P0=g RCxlnT0Cxh T0 P0=T0Cxh T0(g RCx) 4.50 )isamonotonousfunctionwhichdecreaseswithheightbecausetheterminthebracketsislessthanone.Thissituationisroughlyrepresentingthepressureintheatmosphereandresultsinatemperaturedecrease.ItcanbeobservedthatCxhasadoublerolewhichcanchangethepressureratio.Equation( 4.50 )canbeapproximatedbytwoapproaches/ideas.Therstapproximationforasmalldistance,h,andthesecondapproximationforasmalltemperaturegradient.ItcanberecalledthatthefollowingexpansionsareP P0=limh>01Cx RCx=1gh0 }| {gh T0Rcorrectionfactorz }| {RgCxg2h2 4.51 )showsthatthersttwotermsarethestandardterms(negativesignisasexpectedi.e.negativedirection).Thecorrectionfactoroccursonlyatthethirdtermwhichisimportantforlargerheights.Itisworthtopointoutthattheabovestatementhasaqualitativemeaningwhenadditionalparameterisadded.However,thiskindofanalysiswillbepresentedinthedimensionalanalysischap-ter P0=limCx>01Cx RCx=egh RT0gh2Cx RT0:::

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Equation( 4.52 )showsthatthecorrectionfactor(lapsecoefcient),Cx,inuencesatonlylargevaluesofheight.Ithastobenotedthattheseequations( 4.51 )and( 4.52 )arenotproperlyrepresentedwithoutthecharacteristicheight.Ithastobeinsertedtomakethephysicalsignicanceclearer.Equation( 4.50 )representsonlythepressureratio.Forengineeringpur-poses,itissometimesimportanttoobtainthedensityratio.Thisrelationshipcanbeobtainedfromcombiningequations( 4.50 )and( 4.45 ).Thesimplestassumptiontocombinetheseequationsisbyassumingtheidealgasmodel,equation( 2.25 ),toyield 0=PT0 P0z }| {1Cxh T0(g RCx)T0 }| {1+Cxh T 4.50 )isstableandifsounderwhatconditions.Supposethatforsomereason,asmallslabofmaterialmovesfromalayeratheight,h,tolayeratheighth+dh(seeFigure 4.7 )Whatcouldhap-pen?Therearetwomainpossibili-tiesone:theslabcouldreturntotheoriginallayerortwo:stayatthenewlayer(orevenmovefurther,higherheights).Therstcaseisreferredtoasthestableconditionandthesecondcasereferredtoastheunstablecondition.Thewholesystemfallsapartanddoesnotstayiftheanalysispredictsunstableconditions.Aweakwindorotherdisturbancescanmaketheunstablesystemtomovetoanewcondition.Thisquestionisdeterminedbythenetforcesactingontheslab.Whethertheseforcesaretowardtheoriginallayerornot.Thetwoforcesthatactontheslabarethegravityforceandthesurroundingspressure(buoyantforces).Clearly,theslabisinequilibriumwithitssurroundingsbeforethemovement(notnecessarilystable).Underequilibrium,thebodyforcesthatactingontheslabareequaltozero.Thatis,thesurroundingspressureforces(buoyancyforces)areequaltogravityforces.Thebuoyancyforcesareproportionaltotheratioofthedensityoftheslabtosurroundinglayerdensity.Thus,thestabilityquestioniswhethertheslabdensityfromlayerh,0(h)undergoingafreeexpansionishigherorlowerthanthedensityofthelayerh+dh.If0(h)>(h+dh)thenthesituationisstable.Theterm0(h)isslabfromlayerhthathadundergonethefreeexpansion.Thereasonthatthefreeexpansionischosentoexplaintheprocessthattheslabundergoeswhenitmovesfromlayerhtolayerh+dhisbecauseitisthe

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4.53 )asfollowing(h+dh) T0(g RCx)1+Cxdh T 2.25 ))0(h+dh) 4.50 )butcanbeapproximatedbyequation( 4.51 )andthus0(h+dh) T(h)R1=k 4.57 )transformedinto0(h+dh) P1=k 4.57 )intaylorseriesresultsin1gdh P1=k=1gdh Pkg22kg22dh2 4.54 )andthenitisexpandedintaylorseriesas(h+dh) T0(g RCx)1+Cxdh T1g PCx

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Thecomparisonoftherighthandtermsofequations( 4.59 )and( 4.58 )pro-videstheconditionstodeterminethestability.Fromamathematicalpointofview,tokeeptheinequalityforasmalldhonlythersttermneedtobecomparedasg Pk>g PCx 4.60 )andusingtheidealgasidentity,ittransformedtoCx kPCx
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4.11 )canbeused(semionedirectionalsituation)whenrisusedasdirectionandthus@P @r=G r2 P=G RTZrrbdr r2 4.63 )resultsinlnP Pb=G RT1 b=P Pb=eG RTrrb 4.65 )demonstratesthatthepressureisreducedwiththedistance.Itcanbenoticedthatforr!rbthepressureisapproachingP!Pb.Thisequationconrmsthatthedensityinouterspaceiszero(1)=0.Asbefore,equation( 4.65 )canbeexpandedintaylorseriesas b=P Pb=standardz }| {1G(rrb) }| {2GRT+G2rb(rrb)2

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possiblesolution.Thus,equation( 4.63 )istransformedintoZPPbdP P=G ZRTZrrbdr r2 b=P Pb=eG ZRTrrb 4.65 )demonstratesthatthepressureisreducedwiththedistance.Itcanbeobservedthatforr!rbthepressureisapproachingP!Pb.Thisequationconrmsthatthedensityinouterspaceiszero(1)=0.Asbeforetaylorseriesforequation( 4.65 )is b=P Pb=standardz }| {1G(rrb) }| {2GZRT+G2rb(rrb)2 4.35 )isusedwiththehydrostaticuidequationresultsin@P @r=0ePP0 r2 4.70 )iseP0P BT=ConstantBTg0

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drr2 dr+4G=0 4.8 )canbetransformedintoadifferentcoordinatesystemwherethemaincoordinateisinthedirectionoftheeffectivegravity.Thus,thepreviousmethodcanbeusedandthereisnoneedtosolvenewthree(ortwo)differentequations.Asbefore,theconstantpressureplaneisperpendiculartothedirectionoftheeffectivegravity.Generallytheaccelerationisdividedintotwocategories:linearandangularandtheywillbediscussedinthisorder. g Example4.2:

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g=5 9:8127:01Themagnitudeoftheeffectiveaccelerationisjgeffj=p 4.10 .Calculatetheshapeofthesurface.Ifthereisaresistancewhatwillbethean-gle?Whathappenwhentheslopeangleisstraight(thecartisdroppingstraightdown)?SOLUTION

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dr=g !2r 4.81 )canbeintegratedasxx0=!2r2

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Noticethattheintegrationconstantwassubstitutedbyx0.Theconstantpres-surewillbealong Example4.4: 4.13 .Calculatethemini-mumforces,F1andF2tomaintainthegateinposition.Assumingthattheatmo-sphericpressurecanbeignored.SOLUTION }| {Pad|{z}dA(`+)Thepressure,PcanbeexpressedasafunctionasthefollowingP=g(`+)sinTheliquidtotalmomentonthegateisM=Zb0g(`+)sinad(`+)

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4.84 )andalsoacenterofarea.TheseconceptshavebeenintroducedinChapter 3 .Severalrepresentedareasforwhichmomentofin-ertiaandcenterofareahavebeentabulatedinChapter 3 .Thesetabulatedvaluescanbeusedtosolvethiskindofproblems.

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4.14 .Thesymmetryisaroundanyaxesparalleltoaxisx.Thetotalforceandmomentthattheliquidextract-ingontheareaneedtobecalcu-lated.First,theforceisF=ZAPdA=Z(Patmos+gh)dA=APatmos+gZ`1`0h()z }| {(+`0)sindA 4.85 )refertostartingpointandendingpointsnottothestartareaandendarea.Theintegralinequation( 4.85 )canbefurtherdevelopedasFtotal=APatmos+gsin0BBB@`0A+xcAz }| {Z`1`0dA1CCCA }| {h())dA

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}| {Z10dA+gsinIx0x0z }| {Z102dA 4.90 )canbewritteninmorecompactformas 4.4 canbegeneralizedtosolveanytwoforcesneededtobalancethearea/gate.Considerthegeneralsymmetricalbodyshowningure 4.15 whichhastwoforcesthatbalancethebody.Equations( 4.87 )and( 4.91 )canbecombinedthemomentandforceactingonthegeneralarea.Iftheatmosphericpressurecanbezeroorincludeadditionallayerofliquid.TheforcesbalancereadsF1+F2=A[Patmos+gsin(`0+xc)] g(ba) g(ba)

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theretwodifferentmomentsandthereforthreeforcesarerequired.Thus,addi-tionalequationisrequired.Thisequationisfortheadditionalmomentaroundthexaxis(seeforexplanationinFigure 4.16 ).Themomentaroundtheyaxisisgivenbyequation( 4.91 )andthetotalforceisgivenby( 4.87 ).Themomentaroundthexaxis(whichwasarbitrarychosen)shouldbeMx=ZAyPdA 4.96 )intoMx=ZAy(Patmos+gsin)dA 4.96 )canbewrittenasMx=PatmosAycz }| {ZAydA+gsinIx0y0z }| {ZAydA 3 4.87 ),( 4.91 )and( 4.99 )providethebaseforsolvinganyproblemforstraightareaun-derpressurewithuniformdensity.Therearemanycombinationsofproblems(e.g.twoforcesandmo-ment)butnogeneralsolutionispro-vided.Exampletoillustratetheuseoftheseequationsisprovided. Example4.5: 4.17 .SOLUTION

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4.87 ),( 4.91 )and( 4.99 ).Themomentofinertiaofthetrian-glearoundxismadeoftwotriangles(asshownintheFigure( 4.17 )fortrian-gle1and2).Triangle1canbecalcu-latedasthemomentofinertiaarounditscenterwhichis`0+2(`1`0)=3.Theheightoftriangle1is(`1`0)anditswidthbandthus,momentofinertiaaboutitscenterisIxx=b(`1`0)3=36.Themomentofinertiafortriangle1aboutyisIxx1=b(`1`0)3 }| {b(`1`0) 3x12z }| {`0+2(`1`0) 32Theheightofthetriangle2isa(`1`0)anditswidthbandthus,themomentofinertiaaboutitscenterisIxx2=b[a(`1`0)]3 }| {b[a(`1`0)] 3x22z }| {`1+[a(`1`0)] 32andthetotalmomentofinertiaIxx=Ixx1+Ixx2Theproductofinertiaofthetrianglecanbeobtainbyintegration.Itcanbenoticedthatupperlineofthetriangleisy=(`1`0)x b+`0.Thelowerlineofthetriangleisy=(`1`0a)x b+`0+a.Ixy=Zb024Z(`1`0a)x b+`0+a(`1`0)x b+`0xydx35dy=2ab2`1+2ab2`0+a2b2 }| {ab

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4.91 )andequation( 4.99 ).ThepressurecenteristhedistancethatwillcreatethemomentwiththehydrostaticforceonpointO.Thus,thepressurecenterinthexdirectionisxp=1 3.15 )resultsinxp=Ixx

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}| {ZA1dA+2xc2A2z }| {ZA2dA++nxcnAnz }| {ZAndA37775

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Aftersimilarseparationofthetotalintegral,onecanndthatMy=gsinnXi=1iIx0x0i Example4.6: 4.18 .Thelastlayerismadeofwaterwithden-sityof1000[kg=m3].Thedensitiesare1=500[kg=m3],2=800[kg=m3],3=850[kg=m3],and4=1000[kg=m3].Calculatetheforcesatpointsa1andb1.As-sumethatthelayersarestableswithoutanymovementbetweentheliquids.Alsoneglectallmasstransferphenomenathatmayoccur.Theheightsare:h1=1[m],h2=2[m],h3=3[m],andh4=4[m].Theforcesdistancesarea1=1:5[m],a2=1:75[m],andb1=4:5[m].Theangleofinclinationisis=45. 4.114 )and( 4.111 ).Thesolutionmethodofthisexampleisappliedforcaseswithlesslayers(forex-amplebysettingthespecicheightdifferencetobezero).Equation( 4.114 )canbeusedbymodifyingit,ascanbeno-ticedthatinsteadofusingtheregularatmosphericpressurethenewatmosphericpressurecanbeusedasPatmos0=Patmos+1gh1

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}| {`(b2a2)+gsin3Xi=1i+1xciAiThesecondequationis( 4.114 )tobewrittenforthemomentaroundthepointOasF1a1+F2b1=Patmos0xcAtotalz }| {(b2+a2) 2`(b2a2)+gsin3Xi=1i+1Ix0x0iThesolutionfortheaboveequationisF1=2b1gsinP3i=1i+1xciAi2gsinP3i=1i+1Ix0x0i

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4.118 ))itcanbeobservedthattheforceinthedi-rectionofy,forexample,issimplytheintegraloftheareaperpendiculartoyasFy=ZAPdAy

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4.120 )implicitlymeansthatthenetforceonthebodyiszdirectionisonlytheactualliquidaboveit.Forexam-ple,Figure 4.20 showsaoatingbodywithcutoutslotintoit.Theatmo-sphericpressureactsontheareawithcontinuouslines.Insidetheslot,theat-mosphericpressurewithitpiezometricpressureiscanceledbytheupperpartoftheslot.Thus,onlythenetforceistheactualliquidintheslotwhichisact-ingonthebody.Additionalpointthatisworthmentioningisthatthedepthwherethecutoutoccurisinsignicant(neglectingthechangeinthedensity). Example4.7: 4.21 )).Thedamismadeofanarcwiththeangleof0=45andradiusofr=2[m].Youcanassumethattheliquiddensityisconstantandequalto1000[kg=m3].Thegravityis9.8[m=sec2]andwidthofthedameisb=4[m].Comparethedifferentmethodsofcomputations,directandindirect.SOLUTION }| {rcosd

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pressureisonlyafunctionofanditisP=Patmos+grsinTheforcethatisactingonthexdirectionofthedamisAxP.WhentheareaAxisbrdcos.Theatmosphericpressuredoescancelitself(atleastiftheatmosphericpressureonbothsidesofthedamisthesame.).ThenetforcewillbeFx=Z00Pz }| {grsindAxz }| {brcosd 4.22 )asFx=gAxz }| {brsin0xcz }| {rsin0 2sin(45)=19600:0[N]Sincethelasttwoequationsareiden-tical(usethesinuoustheoremtoproveitsin2+cos2=1),clearlythediscussionearlierwasright(notagoodproofLOL }| {0BBB@Az }| {0r2

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4.22 )shouldbecalculatedasyc=ycAarcycAtriangle 4.23 )isatycarc=4rsin2 3.1 and 3.2 .andsubstitutingtheproperval-uesresultsinycr=Aarcz}|{r2 }| {4rsin }| {2rcos }| {sinr2 {z }Aarcr2sincos {z }AtriangleThisvalueisthereversevalueanditisycr=1:65174[m]TheresultofthearccenterfrompointO(abovecalculationarea)isyc=rycr=21:651740:348[m]ThemomentisMv=ycFy0:34822375:27792:31759[Nm]Thecenterpressureforxareaisxp=xc+Ixx }| {b(rcos0)3 {z }xcb(rcos0)=5rcos0

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ThetotalmomentisthecombinationofthetwoanditisMtotal=23191:5[Nm]FordirectintegrationofthemomentisdoneasfollowingdF=PdA=Z00gsinbrdandelementmomentisdM=dF`=dF`z }| {2rsin 4TheverticalforcecanbeobtainedbyFv=Z00PdAvorFv=Z00Pz }| {grsindAvz }| {rdcos=gr2

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4.25 ,calculatethemomentaroundpointOandtheforcecreatedbytheliquidperunitdepth.Thefunctionofthedamshapeisy=Pni=1aixianditisamonotonousfunction(thisrestrictioncanberelaxedsomewhat).Alsocalculatethehorizontalandverticalforces.SOLUTION 4.25 )inthiscaseisdF=Pz }| {hz }| {(by)gdAz }| {p 4.25 ).Itcanbenoticedthatthedifferentialareathatisusedhereshouldbemultipliedbythedepth.Frommathematics,itcanbeshownthatp dx2 dx2=vuut 4.26 ).ThedistancebetweenthepointonthedamatxtothepointOis`(x)=p

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TheanglebetweentheforceandthedistancetopointOis(x)=tan1dy dxtan1by xbxTheelementmomentinthiscaseisdM=`(x)dFz }| {(by)gs dx2cos(x)dxTomakethisexamplelessabstract,considerthespeciccaseofy=2x6.Inthiscase,onlyonetermisprovidedandxbcanbecalculatedasfollowingxb=6r dx=12x5andthederivativeisdimensionless(adimensionlessnumber).Thedistanceis`=vuut 6

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4.27 .Theforcetoholdthecylinderattheplacemustbemadeofintegrationofthepressurearoundthesurfaceofthesquareandcylinderbodies.Theforcesonsquaregeometrybodyaremadeonlyofverticalforcesbecausethetwosidescanceleachother.However,ontheverticaldirection,thepressureonthetwosurfacesaredifferent.Ontheuppersurfacethepressureisg(h0a=2).Onthelowersurfacethepressureisg(h0+a=2).Theforceduetotheliquidpressureperunitdepth(intothepage)isF=g((h0a=2)(h0+a=2))`b=gab`=gV 4.125 )tobeF V=g

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arecanceled.Anybodythathasaprojectedareathathastwosides,thosewillcanceleachother.Anotherwaytolookatthispointisbyapproximation.Foranytworectanglebodies,thehorizontalforcesarecancelingeachother.Thuseventhesebodiesareincontactwitheachother,theimaginarypressuremakeitsothattheycanceleachother.Ontheotherhand,anyshapeismadeofmanysmallrectangles.Theforceoneveryrectangularshapeismadeofitsweightofthevolume.Thus,thetotalforceismadeofthesumofallthesmallrectangleswhichistheweightofthesumofallvolume. 4.27 .Theforceperarea(seeFigure 4.28 )isdF=Pz }| {g(h0rsin)dAverticalz }| {sinrd 4.127 )F=Z20g(h0rsin)rdsin 4.127 )transformsittoF=rgZ20(h0rsin)sind 4.129 )isF=r2g Example4.9: 4.29 ,isoatinginliquidwithdensity,l.Thebodywasinsertedintoliquidinasuchawaythattheairhadremainedinit.Ex-pressthemaximumwallthickness,t,asafunctionofthedensityofthewall,sliquiddensity,landthesurroundingsairtemperature,T1forthebodytooat.Inthecasewherethicknessishalfthemaximum,calculatethepressureinsidethecontainer.Thecontainerdiameterisw.Assumethatthewallthicknessissmallcomparedwiththeotherdimensions(t<
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}| {w2hairz }| {Patmos }| {w2+2wh1CAtsTheliquidamountentersintothecavityissuchthattheairpressureinthecavityequalstothepressureattheinterface(inthecavity).Notethatforthemaximumthickness,theheight,h1hastobezero.Thus,thepressureattheinterfacecanbewrittenasPin=lghinOntheotherhand,thepressureattheinterfacefromtheairpointofview(idealgasmodel)shouldbePin=mairRT1 {z }VSincetheairmassdidn'tchangeanditisknown,itcanbeinsertedintotheaboveequation.lghin+Patmos=Pin=w2hz }| {Patmos

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Example4.10: Patmos+2h3l2g2 }| {w2(hhin)g=containerz }| {w2+2whtsg+airz }| {w2hPatmos

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}| {Vglbodyweightz }| {Vgl=Vgl(1)ButontheothersidetheinternalforceisF=ma=mz }| {Vla

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Thus,theaccelerationisa=g1 Iftheobjectisleftatrest(nomovement)thustimewillbe(h=1=2at2)t=s g(1)Iftheobjectisverylight(!0)thentmin=s g+p 2 2 2 Example4.12: }| {rcoscosdAxz }| {coscosdAz }| {r2ddThetotalforceisthenFx=Z0Z0(LS)gcos2cos2r3ddTheresultoftheintegrationtheforceonsphereisFs=2(LS)r3

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r= 4.30 showsabodymadeofhollowballoonandaheavysphereconnectedbyathinandlightrod.Thisarrangementhasmasscen-troidclosetothemiddleofthesphere.Thebuoyantcenterisbe-lowthemiddleoftheballoon.Ifthisarrangementisinsertedintoliquidanditwillbeoating,theballoonwillbeonthetopandsphereonthebottom.Tiltingthebodywithasmallanglefromitsrestingpo-sitioncreatesashiftintheforcesdirection(examineFigure 4.30 b).Theseforcescreateamomentwhichwantstoreturnthebodytotheresting(orig-inal)position.WhenthebodyisatthepositionshowninFigure 4.30 c,thebodyisunstableandanytiltfromtheoriginalpositioncreatesmomentthatwillfurthercontinuetomovethebodyfromitsoriginalposition.Thisanalysisdoesn'tviolatethesecondlawofthermodynamics.Movingbodiesfromanunstablepositionisinessencelikeapotential. 4.31 ).ThissituationissimilartoFigure 4.30 c.However,anyexperimentofthiscubicwoodshowsthatitisstablelo-cally.Smallamountoftiltingofthecubicresultsinreturningtotheoriginalposition.Whentiltingalargeramountthan=4,itresultsinaippingintothenextstable

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position.Thecubicisstableinsixpositions(everycubichassixfaces).Infact,inanyofthesesixpositions,thebodyisinsituationlikein 4.30 c.Thereasonforthislocalstabilityofthecubicisthatotherpositionsarelessstable.Ifonedrawsthestability(lateraboutthiscriterion)asafunctionoftherotationanglewillshowasinusoidalfunctionwithfourpicksinawholerotation. 4.32 .Thecenterofthemass(gravity)isstillintheoldlocationsincethebodydidnotchange.Thestabilityofthebodyisdividedintothreecate-gories.Ifthenewimmersevolumecreatesanewcenterinsuchwaythatthecoupleforces(gravityandbuoyancy)trytoreturnthebody,theoriginalstateisre-ferredasthestablebodyandviceversa.Thethirdstateiswhenthecoupleforcesdohavezeromoment,itisreferredtoastheneutralstable.Thebody,showninFigure 4.32 ,whengivenatiltedposition,movetoanewbuoyantcenter,B'.Thisdeviationofthebuoyantcenterfromtheoldbuoyantcenterlocation,B,shouldtobecalculated.Thisanalysisisbasedonthedifferenceofthedisplacedliquid.Therightgreenarea(volume)inFigure 4.32 isdisplacedbythesamearea(reallythevolume)onleftsincetheweightofthebodydidn'tchange 4.32 asF.Thedisplacementofthebuoyantcentercanbecalculatedbyexaminingthemomenttheseforcesarecreating.Thebodyweightcreatesoppositemomenttobalancethemomentofthedisplacedliquidvolume.

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}| {glxdA| {z }dVx=glZAx2dA 4.134 )isreferredtoastheareamomentofinertiaandwasdiscussedinChapter 3 .Thedistance, 4.134 )as 4.135 )with( 4.136 )yields 1 4.32 ,thegeometricalquantitiescanberelatedas }| {lIxx Example4.13:

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4.139 )requiresthatseveralquantitiesshouldbeexpressed.Themo-mentofinertiaforablockisgiveninTable 3.1 andisIxx=La3 }| {ahL=limmersedvolumez }| {ah1L=)h1=s 4.31 ) }| {s 2=h }| {La3 {z }Vh h=1 12a h21 2(1) hp 4.142 )canbeexpressed.Forcylinder(circle)themomentofinertiaisIxx=b4=64.Thedistance 4.140 )).Thus,theequationis h=g h21 2(1)

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hp 4.35 }| {p Example4.14:

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4.36 ).AbodyisloadedwithliquidBandisoatinginaliquidAasshowninFigure 4.36 .Whenthebodyisgivenatiltingpositionthebodydisplacestheliquidontheoutside.Atthesametime,theliquidinsideischangingitsmasscentroid.ThemomentcreatedbytheinsidedisplacedliquidisMin=glBIxxB

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{z } 4.144 )showsthat WtotalnXi=1 WtotalnXi=1Ixxbi

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Wtotal 4.152 )providesthesolution.Thecalculationof 4.153 )withequation( 4.152 )resultsin mship

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4.38 .Thebodyweightdoesn'tchangeduringtherotationthatthegreenareaontheleftandthegreenareaonrightarethesame(seeFigure 4.38 ).Therearetwosituationsthatcanoccur.Afterthetilt-ing,theupperpartofthebodyisabovetheliquidorpartofthebodyissub-mergedunderthewater.Themathe-maticalconditionfortheborderiswhenb=3a.Forthecaseofb<3athecalculationofmomentofinertiaaresimilartothepreviouscase.Themomentcreatedbychangeinthedisplacedliquid(area)actinthesamefashionasthebefore.Thecenterofthemomentisneededbefound.Thispointistheintersectionoftheliquidlinewiththebrownmiddleline.Themomentofinertiashouldbecalculatedaroundthisaxis.Forthecasewhereb<3axsomepartisundertheliquid.Theamountofareaundertheliquidsectiondependsonthetiltingangle.Thesecalculationsaredoneasifnoneofthebodyundertheliquid.Thispointisintersectionpointliquidwithlowerbodyanditisneededtobecalculated.Themomentofinertiaiscalculatedaroundthispoint(notethebodyisendedatendoftheupperbody).However,themomenttoreturnthebodyislargerthanactuallywascalculatedandthebodiestendtobemorestable(alsoforotherreasons). 2p `whichmeasuredinHz.

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Theperiodofthecycleis2p }| {Vs 2s Ibody Example4.15:

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4.3.3.2 )arealwaysstablebutunstableofthedensityisinthereversedorder.Supposedthataliquiddensityisarbitraryfunctionoftheheight.Thisdistor-tioncanbeasaresultofheavyuidabovethelighterliquid.Thisanalysisasksthequestionwhathappenwhenasmallamountofliquidfromabovelayerenterintothelowerlayer?Whetherthisliquidcontinueandwillgroworwillitreturntoitsoriginalconditions?Thesurfacetensionistheoppositemechanismthatwillreturnstheliquidtoitsoriginalplace.Thisanalysisisreferredtothecaseofinniteorverylargesurface.Thesimpliedcaseisthetwodifferentuniformdensities.Forexampleaheavyuiddensity,L,aboveloweruidwithlowerdensity,G.Forperfectlystraightinterface,theheavyuidwillstayabovethelighteruid.Ifthesurfacewilldisturbed,someofheavyliquidmovesdown.Thisdisturbancecangroworreturnedtoitsoriginalsituation.Thisconditionisdeterminedbycom-petingforces,thesurfacedensity,andthebuoyancyforces.Theuidabovethedepressionisinequilibriumwiththesoundingpressuresincethematerialisex-tendingtoinnity.Thus,theforcethatactingtogettheaboveuiddownisthebuoyancyforceoftheuidinthedepression.

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(surfacetension)arenotsufcient,thesituationisunstableandtheheavyliq-uidentersintotheliquiduidzoneandviceversa.Asusualthereistheneutralstablewhentheforcesareequal.Anycon-tinuesfunctioncanbeexpandedinseriousofcosines.Thus,exampleofacosinefunctionwillbeexamined.Theconditionsthatrequiredfromthisfunctionwillberequiredfromalltheotherfunctions.Thedisturbanceisofthefollowingh=hmaxcos2x L 4.40 ).Thus,ifthecenterpointofthedepressioncanholdtheintrusiveuidthenthewholesystemisstable.Theradiusofanyequationisexpressedbyequation( 1.47 ).Therstderiva-tiveofcosaroundzeroissinwhichisapproachingzeroorequaltozero.Thus,equation( 1.47 )canbeapproximatedas1 dx2 1.38 )thepressuredifferenceorthepressurejumpisduetothesurfacetensionatthispointmustbePHPL=4hmax2 4.162 )and( 4.163 )showthatiftherelationshipis42 g(HL)

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4.41 .Ifalltheheavyliquidattemptstomovestraightdown,thelighterliquidwillpreventit.Thelighterliquidneedstomoveupatthesametimebutinadifferentplace.Theheavierliquidneedstomoveinonesideandthelighterliquidinanotherlocation.Inthisprocesstheheavierliq-uidenterthelighterliquidinonepointandcreatesadepressionasshowninFigure 4.41 4.41 .Therstcontrolvol-umeismadeofacylinderwithara-diusrandthesecondisthedepres-sionbelowit.Theextralinesofthedepressionshouldbeignored,theyarenotpartofthecontrolvolume.Thehori-zontalforcesaroundthecontrolvolumearecancelingeachother.Atthetop,theforceisatmosphericpressuretimesthearea.Atthecylinderbottom,theforceisghA.Thisactsagainstthegrav-ityforcewhichmakethecylindertobeinequilibriumwithitssurroundingsifthepressureatbottomisindeedgh.Forthedepression,theforceatthetopisthesameforceatthebottomofthecylinder.Atthebottom,theforceistheintegralaroundthedepression.Itcanbeapproximatedasaatcylinderthathasdepthofr=4(readtheexplanationintheexample 4.12 )Thisvalueisexactiftheshapeisaperfecthalfsphere.Inreality,theerrorisnotsignicant.Additionallywhenthedepressionoccurs,theliquidlevelisreducedabitandthelighterliquidisllingthemissingportion.Thus,theforceatthebottomisFbottomr2hr 4.41 ,thetotalforceisthenF=2rcos

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Theradiusisobtainedbyrs 4.42 ).Thisradiusislimitedbecausethelighterliquidhastoenteratthesametimeintotheheavierliquidzone.Sincetheexchangevolumesofthesetwoprocessarethesame,thespecicradiusislimited.Thus,itcanbewrittenthattheminimumradiusisrmintube=2s g(LG) 4.172 .ThisanalysisintroducesanewdimensionalnumberthatwillbediscussedinagreaterlengthintheDimensionlesschapter.Inequation( 4.172 )theanglewasassumedtobe90degrees.However,thisangleisneverobtained.Theactualvalueofthisangleisabout=4to=3andinonlyextremecasestheangleexceedthisvalue(consideringdynamics).InFigure 4.42 ,itwasshownthatthedepressionandtheraisedareaarethesame.Theactualareaofthedepression

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Example4.17: 4.171 )canbeused.Thedensityofairisnegligibleascanbeseenfromthetemperaturecomparetothealuminumdensity.rvuut 24009:81Theminimumradiusisr0:02[m]whichdemonstratestheassumptionofh>>rwasappropriate.

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Ingeneral,theco-currentisthemorecommon.Additionally,thecountercurrentowmusthavespecialcongurationsoflonglengthofow.Generally,thecountercurrentowhasalimitedlengthwindowofpossibilityinaverticalowinconduitswiththeexceptionofmagnetohydrodynamics.Theowregimesarereferredtothearrangementoftheuids.Themaindifferencebetweentheliquidliquidowtogas-liquidowisthatgasdensityisextremelylighterthantheliquiddensity.Forexample,waterandairowasopposetowaterandoilow.Theothercharacteristicthatisdifferentbetweenthegasowandtheliquidowisthevariationofthedensity.Forexample,areductionofthepressurebyhalfwilldoublethegasvolumetricowratewhilethechangeintheliquidisnegligible.Thus,theowofgasliquidcanhaveseveralowregimesinonesituationwhiletheowofliquidliquidwill(probably)haveonlyoneowregime. 5.2 .Thiskindofowregimeisreferredtoashorizontalow.Whentheowrateofthelighterliquidisalmostzero,theowisreferredtoasopenchannelow.Thisdenition(openchannelow)continuesforsmallamountoflighterliq-uidaslongastheheavierowcanbecalculatedasopenchannelow(ignoringthelighterliquid).Thegeometries(eventheboundaries)ofopenchannelowareverydiverse.Openchannelowappearsinmanynature(river)aswellinindustrial

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5.3 ).Thewaveshapeiscreatedtokeepthegasandtheliquidvelocityequalandatthesametimetohaveshearstresstobebalancebysurfacetension.Thecongura-tionofthecrosssectionnotonlydependonthesurfacetension,andotherphysicalpropertiesoftheuidsbutalsoonthematerialoftheconduit. 5.4 ).Thepressuredropofthiskindofregimeissignicantlylargerthanthestratiedow.Theslugowcannotbeassumedtobeashomogeneousownoritcanexhibitsomeaverageviscosity.Theaverageviscositydependsontheowandthusmakingitasinsignicantwaytodothecalculations.Furtherincreaseofthelighterliquidowratemovetheowregimeintoannularow.Thus,thepossibilitytogothroughslugowregimedependsonifthereisenoughliquidowrate.

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cursrelativelycloser/sooner.Thus,theowthatstartsasastratiedowwillturnedintoaslugoworstratiedwavy 5.5 .Asmanythingsinmultiphase,thismapisonlycharacteristicsofthenormalconditions,e.g.innormalgravitation,weaktostrongsurfacetensioneffects(air/waterinnormal

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5.6 ).Notice,thedifferentmechanismincreatingtheplugowinhorizontalowcomparedtotheverticalow.Furtherincreaseoflighterliquidowratewillincreasetheslugsizeasmorebubblescollidetocreatesuperslug;theowregimeisreferredaselongatedbubbleow.Theowislessstableasmoreturbulentowandseveralsuper

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slugorchurnowappearsinmorechaoticway,seeFigure 5.6 .Afteradditionalincreaseofsuperslug,alltheseelongatedslugunitetobecomeanannularow.Again,itcanbenotedthedifferenceinthemechanismthatcreateannularowforverticalandhorizontalow.Anyfurtherincreasetransformstheouterliquidlayerintobubblesintheinnerliquid.Flowofnearverticalagainstthegravityintwophasedoesnotdeviatefromvertical.Thechokingcanoccuratanypointdependsontheuidsandtemperatureandpressure. 5.7 ).Intheliterature,Figure 5.7 presentedindimensionlesscoordinates.TheabscissaisafunctionofcombinationofFroude,Reynolds,andWebernumbers.Theordinateisacombinationofowrateratioanddensityratio.FlowWithTheGravityAsopposedtotheowagainstgravity,thisowcanstartswithstratiedow.Agoodexampleforthisowregimeisawaterfall.Theinitialpartforthisowismoresignicant.Sincetheheavyliquidcanbesuppliedfromthewrongpoint/side,theinitialparthasalargersectioncomparedtotheowagainstthegrav-ityow.Aftertheowhassettled,theowcontinuesinastratiedconguration.Thetransitionsbetweentheowregimesissimilartostratiedow.However,the

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A

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andfortheliquidQL=GL GA=QG LA=QL

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Q 5.1 )and( 5.7 )intoequation( 5.16 )resultsinaverage=_mGz }| {X_m+_mLz }| {(1X)_m QG+QL=X_m+(1X)_m X_m G| {z }QG+(1X)_m L| {z }QL 5.17 )canbesimpliedbycancelingthe_mandnoticingthe(1X)+X=1tobecome G+(1X) G+(1X)

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TherelationshipbetweenXandisX=_mG LULA(1)| {z }AL+GUGA=GUG LUL(1)+GUG 5.20 )becomesX=G L(1)+G 5.13 ))isUm=QL+QG Example5.1: 5.23 )iscorrect?SOLUTION Thegoverningmomentumequationcanbeapproximatedas_mdUm dxSwAmgsin

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5.24 )as dx=S Aw_m AdUm dxdw dx=_md dxhm+Um2 5.25 ),thetotalpressurelosscanbewrittenas dx=frictionz }| {dP dxf+accelerationz }| {dP dxa+gravityz }| {dP dxg dxf=S Aw dxf=4w

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Thefrictionfactorismeasuredforasinglephaseowwheretheaveragevelocityisdirectlyrelatedtothewallshearstress.Thereisnotavailableexperimentaldatafortherelationshipoftheaveragedvelocityofthetwo(ormore)phasesandwallshearstress.Infact,thisfrictionfactorwasnotmeasuredfortheaveragedviscosityofthetwophaseow.Yet,sincethereisn'tanythingbetter,theexperimentaldatathatwasdevelopedandmeasuredforsingleowisused.Thefrictionfactorisobtainedbyusingthecorrelationf=CmUmD mn 5.33 )averageviscosityasaverage=1 G+(1X) dxa=_mdUm 5.35 )canbewrittenasdP dxa=_md dx_m Am! 5.36 )becomesdP dxa=_m2266664pressurelossduetodensitychangez }| {1 dx1 }| {1 dx377775

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4 andisdP dxg=gmsin 5.38 )isthedensitywithoutthemovement(thestaticdensity). dxdx }| {Pabf+accelerationz }| {Paba+gravityz }| {Pabg dxTPdP dxSGf

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WheretheTPdenotesthetwophasesandSGdenotesthepressurelossforthesinglegasphase.EquivalentdenitionfortheliquidsideisL=s dxTPdP dxSLf dxSLdP dxSGf dxSG=dP dxSL dxL=2fLUL2l dxG=2fGUG2l

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Theowwiththegravityandlighterdensitysolidparticles. 2. Theowwiththegravityandheavierdensitysolidparticles. 3. Theowagainstthegravityandlighterdensitysolidparticles. 4. Theowagainstthegravityandheavierdensitysolidparticles.Allthesepossibilitiesaredifferent.However,therearetwosetsofsimilarcharacteristics,possibility,1and4andthesecondsetis2and3.Therstsetissimilarbecausethesolidparticlesaremovingfasterthantheliquidvelocityandviceversaforthesecondset(slowerthantheliquid).Thediscussionhereisaboutthelastcase(4)becauseverylittleisknownabouttheothercases. }| {D3g(SL) 6=dragforcesz }| {CD1D2LUL2 5.48 )intoequation( 5.47 )becomeCD1(UL)z }| {f(Re)UL2=4Dg(SL) 3L 5.49 )relatestheliquidvelocitythatneededtomaintaintheparticleoatingtotheliquidandparticlesproperties.Thedragcoefcient,CD1iscom-plicatedfunctionoftheReynoldsnumber.However,itcanbeapproximatedforseveralregimes.TherstregimeisforRe<1whereStokes'Lawcanbeapproxi-matedasCD1=24

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Intransitionalregion1
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5.9 ).

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microgravity).Therewasverylittleinvestigationsandknownaboutthesolidliquidowingdown(withthegravity).Furthermore,thereisverylittleknowledgeaboutthesolidliquidwhenthesoliddensityissmallerthantheliquiddensity.Thereisnoknownowmapforthiskindofowthatthisauthorisawareof.Nevertheless,severalconclusionsand/orexpectationscanbedrawn.Theissueofminimumterminalvelocityisnotexistandthereforthereisnoxedormixeduidizedbed.Theowisfullyuidizedforanyliquidowrate.TheowcanhaveslugowbutmorelikelywillbeinfastFluidizationregime.Theforcesthatactonthesphericalparticlearethebuoyancyforceanddragforce.ThebuoyancyisacceleratingtheparticleanddragforcearereducingthespeedasD3g(SL) 6=CD1D2L(USUL)2 5.54 ,itcanobservedthatincreaseoftheliquidvelocitywillincreasethesolidparticlevelocityatthesameamount.Thus,forlargevelocityoftheuiditcanbeobservedthatUL=US!1.However,forasmalluidvelocitythevelocityratioisverylarge,UL=US!0.Theaffectivebodyforceseemsbytheparticlescanbeinsomecaseslargerthanthegravity.Theowregimeswillbesimilarbutthetransitionwillbeindifferentpoints.Thesolidliquidhorizontalowhassomesimilaritytohorizontalgasliquidow.Initiallythesolidparticleswillbecarriedbytheliquidtothetop.Whentheliquidvelocityincreaseandbecameturbulent,someoftheparticlesenterintotheliquidcore.Furtherincreaseoftheliquidvelocityappearassomewhatsimilartoslugow.However,thisauthorhavenotseenanyevidencethatshowtheannularowdoesnotappearinsolidliquidow.

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5.10 depictsemptyingofcanlledwithliq-uid.Theairisattemptingtoenterthecavitytollthevacuumcreatedthusforcingpulseow.Iftherearetwoholes,insomecases,liquidowsthroughoneholeandtheairthroughthesecondholeandtheowwillbecontinuous.Italsocanbenoticedthatifthereisonehole(orice)andalongandnarrowtube,theliquidwillstayinthecavity(neglectingotherphenom-enasuchasdrippingow.). Therearethreeowregimes 5.12 ).Thenamepulseowisusedtosignifythattheowisowinginpulsesthatoccursinacertainfrequency.Thisisopposedtocountercurrentsolidgasowwhenalmostnopulsewasobserved.Initially,duetothegravity,theheavyliquidisleavingthecan.Thenthepressureinthecanisreducedcomparedtotheoutsideandsomelighterliquid(gas)enteredintothecan.Then,thepressureinthecanincrease,andsomeheavyliquidwillstartstoow.Thisprocesscontinueuntilalmosttheliquidisevacuated(someliquidstayduethesur-facetension).Inmanysituations,thevolumeowrateofthetwophaseisalmostequal.Thedurationthecycledependsonseveralfactors.Thecycledurationcanbereplacedbyfrequency.Theanalysisofthefrequencyismuchmorecomplexissueandwillnotbedealthere.

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4.7 ).Theratioofthediametertothelengthwithsomecombinationsofthephysicalproperties(surfacetensionetc)determinesthepointwherethecounterowcanstart.Atthispoint,thepulsingowwillstartandlargerdiameterwillincreasetheowandturntheowintoannularow.Ad-ditionalincreaseofthediameterwillchangetheowregimeintoextendedopenchannelow.Extendedopenchannelowretainsthecharacteristicofopenchan-nelthatthelighterliquid(almost)doesnoteffecttheheavierliquidow.Exampleofsuchowinthenatureiswaterfallsinwhichwaterowsdownandair(wind)owsup.Thedrivingforceisthesecondparameterwhicheffectstheowexistence.Whenthedriving(body)forceisverysmall,nocountercurrentowispossible.Considerthecaninzerogravityeld,nocountercurrentowpossible.However,ifthecanwasonthesun(ignoringtheheattransferissue),theowregimeinthecanmovesfrompulsetoannularow.Furtherincreaseofthebodyforcewillmove

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4.7 )page 108 )itcanbeconsideredsta-bleforsmalldiameters.Aowinaverynar-rowtubewithheavyuidabovethelighteruidshouldbeconsideredasaseparateissue.Whentheowrateofbothuidsisverysmall,theowwillbestratiedcountercurrentow.Theowwillchangetopulseowwhentheheavyliquidowrateincreases.Furtherincreaseoftheowwillresultinasinglephaseowregime.Thus,closingthewindowofthiskindofow.Thus,thisincreaseterminatesthetwophaseowpossibility.TheowmapofthehorizontalowisdifferentfromtheverticalowandisshowninFigure 5.14 .Aowinanangleofinclinationisclosertoverticalowunlesstheangleofinclinationisverysmall.Thestratiedcounterowhasalowerpressureloss(fortheliquidside).Thechangetopulseowincreasesthepressurelossdramatically. 5.13 .Theliquidvelocityatverylowgasvelocityisconstantbutnotuniform.Furtherincreaseofthegasvelocitywillreducetheaverageliquidvelocity.Additionalincreaseofthegasvelocitywillbringittoapointwheretheliquidwillowinareversedirectionand/ordisappear(driedout).

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5.15 .Itisassumedthatbothuidsareowinginalaminarregimeandsteadystate.Additionally,itisassumedthattheentranceef-fectscanbeneglected.Theliquidowrate,QL,isunknown.However,thepressuredifferenceinthe(xdirection)isknownandequaltozero.Theboundaryconditionsfortheliquidisthatvelocityatthewalliszeroandthevelocityattheinter-faceisthesameforbothphasesUG=ULorijG=ijL.Asitwillbeshownlater,bothcon-ditionscannotcoexist.Themodelcanbeim-provedbyconsideringturbulence,masstrans-fer,wavyinterface,etc 5.55 )resultsinxy=Lgx+C1 5.58 )toobtainedLdUy

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5.60 )canbeintegratetoyieldUy=Lg Lx2 L+C2 Lx2 L LLgh2 Lx2 L 5.65 )isxj@UL=0=2h2i Lh2 L

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Theowratecanbecalculatedbyintegratingthevelocityacrosstheentireliquidthicknessofthelm.Q w=Zh0Uydx=Zh0Lg Lx2 Ldx 5.15 ).Integrationequation( 5.68 )resultsinQ w=h2(3i2ghL) 6L 5.69 )isequatedtozero.Therearethreesolutionsforequation( 5.69 ).Thersttwosolutionsareidenticalinwhichthelmheightish=0andtheliquidowrateiszero.But,also,theowrateiszerowhen3i=2ghL.Thisrequestisidenticaltothedemandinwhich 5.67 )).Forthisshearstress,thecriticalupwardinterfacevelocityis 31 2)z}|{1 6Lgh2 dxx=0=L0BB@Lg L*02xh+iz }| {2ghL 5.72 ) 5.70 )hastobeequalghLtosupporttheweightoftheliquid.

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dy 5.74 )canberewrittenasdxyG L GL 5.76 )canbeintegratedtwicetoyieldUG=P GLx2+C1x+C2 5.78 )intoequation( 5.77 )resultsinUG=0=P GLD2+C1D+C2 GLD2+C1D

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WhichleadstoUG=P GLx2D2+C1(xD) 5.79 )(a),becomesLgh2 GLh2D2+C1(hD) GLx2D2+Lgh2(xD) 6L(hD)P(h+D)(xD) 5.84 )isequaltothevelocityequation( 5.63 )when(x=h).However,inthatcase,itiseasytoshowthatthegasshearstressisnotequaltotheliquidshearstressattheinterface(whenthevelocitiesareassumedtobetheequal).Thedifferenceinshearstressesattheinterfaceduetothisassumption,oftheequalvelocities,causethisassumptiontobenotphysical.Thesecondchoiceistousetheequalshearstressesattheinterface,condi-tion( 5.79 )(b).ThisconditionrequiresthatGdUG }| {2hP L+GC1=liquidsidez }| {2ghL GL GLx2D2+2ghL GL(xD)

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GLh2D2+2ghL GL(hD) GL+2ghL GLx=D 5.16 describesthegeneralforcesthatactsonthecontrolvolume.Therearetwo

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forcesthatactagainstthegravityandtwoforceswiththegravity.Thegravityforceonthegascanbeneglectedinmostcases.ThegravityforceontheliquidistheliquidvolumetimestheliquidvolumeasFgL=gVolme=wz}|{hL 5.16 )FgL+A=wz}|{LwG=A=wz}|{LwL+forceduetopressurez }| {DP 5.93 )resultingLh+L2P(Dh) 5.94 )resultsin4gLh 5.96 )indicatesthatwhenD>2hisaspecialcase(extendopenchannelow).

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Horizontalow,117Hydrostaticpressure,55,88IIdealgas,61Kkinematicviscosity,11LLapserate,69Linearacceleration,72Liquidphase,63LiquidLiquidRegimes,116Lockhartmartinellimodel,128MMagnicationfactor,61Massvelocity,122Metacentricpoint,100Minimumvelocitysolidliquidow,130Mixeduidizedbed,131Multiphaseow,113Multiphaseowagainstthegravity,120NNeutralmomentZeromoment,99Neutralstable,69,99,109Newtonianuids,1,8OOpenchannelow,117PPendulumaction,106Piezometricpressure,58Pneumaticconveying,132Polynomialfunction,89Pressurecenter,81pseudoplastic,11Pulseow,134

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Verticalcountercurrentow,134Verticalow,117WWatson'smethod,19Westinghousepatent,116