《化工应用数学》

化工應用數學授課教師：郭修伯Lecture6FunctionsanddefiniteintegralsVectors编辑课件Chapter5FunctionsanddefiniteintegralsTherearemanyfunctionsarisinginengineeringwhichcannotbeintegratedanalyticallyintermsofelementaryfunctions.Thevaluesofmanyintegralshavebeentabulated,muchnumericalworkcanbeavoidediftheintegraltobeevaluatedcanbealteredtoaformthatistabulated.Ref.pp.153Wearegoingtostudysomeofthesespecialfunctions…..编辑课件SpecialfunctionsFunctionsDetermineafunctionalrelationshipbetweentwoormorevariablesWehavestudiedmanyelementaryfunctionssuchaspolynomials,powers,logarithms,exponentials,trigonometricandhyperbolicfunctions.FourkindsofBesselfunctionsareusefulforexpressingthesolutionsofaparticularclassofdifferentialequations.Legendrepolynomialsaresolutionsofagroupofdifferentialequations.Learnsomemorenow….编辑课件TheerrorfunctionItoccursinthetheoryofprobability,distributionofresidencetimes,conductionofheat,anddiffusionmatter:0xzerfxz:dummyvariableProofinnextslide编辑课件xandyaretwoindependentCartesiancoordinatesinpolarcoordinatesErrorbetweenthevolumedeterminedbyx-yandr-Thevolumeofhasabaseareawhichislessthan1/2R2andamaximumheightofe-R2

编辑课件MoreabouterrorfunctionDifferentiationoftheerrorfunction:Integrationoftheerrorfunction:Theaboveequationistabulatedunderthesymbol“ierfx”with(Therefore,ierf0=0)Anotherrelatedfunctionisthecomplementaryerrorfunction“erfcx”编辑课件Thegammafunctionforpositivevaluesofn.tisadummyvariablesincethevalueofthedefiniteintegralisindependentoft.(N.B.,ifniszerooranegativeinteger,thegammafunctionbecomesinfinite.)repeatThegammafunctionisthusageneralizedfactorial,forpositiveintegervaluesofn,thegammafunctioncanbereplacedbyafactorial.(Fig.5.3pp.147)编辑课件MoreaboutthegammafunctionEvaluate编辑课件Chapter7VectoranalysisIthasbeenshownthatacomplexnumberconsistedofarealpartandanimaginarypart.Onesymbolwasusedtorepresentacombinationoftwoothersymbols.Itismuchquickertomanipulateasinglesymbolthanthecorrespondingelementaryoperationsontheseparatevariables.Thisistheoriginalideaofvector.Anynumberofvariablescanbegroupedintoasinglesymbolintwoways:(1)Matrices(2)TensorsTheprincipaldifferencebetweentensorsandmatricesisthelabellingandorderingofthemanydistinctparts.编辑课件TensorsGeneralizedaszmAtensoroffirstranksinceonesuffixmisneededtospecifyit.Thenotationofatensorcanbefurthergeneralizedbyusingmorethanonesubscript,thuszmnisatensorofsecondrank(i.e.m,n).Thesymbolismforthegeneraltensorconsistsofamainsymbolsuchaszwithanynumberofassociatedindices.Eachindexisallowedtotakeanyintegervalueuptothechosendimensionsofthesystem.Thenumberofindicesassociatedwiththetensoristhe“rank”ofthetensor.编辑课件Tensorsofzerorank(atensorhasnoindex)Itconsistsofonequantityindependentofthenumberofdimensionsofthesystem.Thevalueofthisquantityisindependentofthecomplexityofthesystemanditpossessesmagnitudeandiscalleda“scalar”.Examples:energy,time,density,mass,specificheat,thermalconductivity,etc.scalarpoint:temperature,concentrationandpressurewhichareallsignedbyanumberwhichmayvarywithpositionbutnotdependupondirection.编辑课件Tensorsoffirstrank(atensorhasasingleindex)Thetensoroffirstrankisalternativelynamesa“vector”.Itconsistsofasmanyelementsasthenumberofdimensionsofthesystem.Forpracticalpurposes,thisnumberisthreeandthetensorhasthreeelementsarenormallycalledcomponents.Vectorshavebothmagnitudeanddirection.Examples:force,velocity,momentum,angularvelocity,etc.编辑课件Tensorsofsecondrank(atensorhastwoindices)Ithasamagnitudeandtwodirectionsassociatedwithit.Theonetensorofsecondrankwhichoccursfrequentlyinengineeringisthestresstensor.Inthreedimensions,thestresstensorconsistsofninequantitieswhichcanbearrangedinamatrixform:编辑课件ThephysicalinterpretationofthestresstensorxzypxxxyxzThefirstsubscriptdenotestheplaneandthesecondsubscriptdenotesthedirectionoftheforce.xyisreadas“theshearforceonthexfacingplaneactingintheydirection”.编辑课件GeometricalapplicationsIfAandBaretwopositionvectors,findtheequationofthestraightlinepassingthroughtheendpointsofAandB.ABC编辑课件ApplicationofvectormethodforstagewiseprocessesInanystagewiseprocess,thereismorethanonepropertytobeconservedandforthepurposeofthisexample,itwillbeassumedthatthethreeproperties,enthalpy(H),totalmassflow(M)andmassflowofonecomponent(C)areconserved.Insteadofconsideringthreeseparatescalarbalances,onevectorbalancecanbetakenbyusingasetofcartesiancoordinatesinthefollowingmanner:UsingxtomeasureM,ytomeasureHandztomeasureCAnyprocessstreamcanberepresentedbyavector:MHCAsecondstreamcanberepresentedby:编辑课件Usingvectoraddition,Thus,ORwithrepresentsofthesumofthetwostreamsmustbeaconstantvectorforthethreepropertiestobeconservedwithinthesystem.Toperformacalculation,wheneitherofthestreamsOMorONisdetermined,theotherisobtainedbysubtractionfromtheconstantOR.Example:whenx=1,Ponchon-Savaritmethod(enthalpy-concentrationdiagram)xyzMRNBAPTheconstantlineORcrosstheplanex=1atpointPOpointAis:pointBis:pointPis:编辑课件MultiplicationofvectorsTwodifferentinteractions(what’sthedifference?)Scalarordotproduct:thecalculationgivingtheworkdonebyaforceduringadisplacementworkandhenceenergyarescalarquantitieswhicharisefromthemultiplicationoftwovectorsifA·B=0ThevectorAiszeroThevectorBiszero=90°AB编辑课件Vectororcrossproduct:nistheunitvectoralongthenormaltotheplanecontainingAandBanditspositivedirectionisdeterminedastheright-handscrewrulethemagnitudeofthevectorproductofAandBisequaltotheareaoftheparallelogramformedbyAandBifthereisaforceFactingatapointPwithpositionvectorrrelativetoanoriginO,themomentofaforceFaboutOisdefinedby:ifAB=0ThevectorAiszeroThevectorBiszero=0°AB编辑课件Commutativelaw:Distributionlaw:Associativelaw:编辑课件UnitvectorrelationshipsItisfrequentlyusefultoresolvevectorsintocomponentsalongtheaxialdirectionsintermsoftheunitvectorsi,j,andk.编辑课件ScalartripleproductThemagnitudeofisthevolumeoftheparallelepipedwithedgesparalleltoA,B,andC.ABCAB编辑课件VectortripleproductThevectorisperpendiculartotheplaneofAandB.WhenthefurthervectorproductwithCistaken,theresultingvectormustbeperpendiculartoandhenceintheplaneofAandB:ABCABwheremandnarescalarconstantstobedetermined.SincethisequationisvalidforanyvectorsA,B,andCLetA=i,B=C=j:编辑课件DifferentiationofvectorsIfavectorrisafunctionofascalarvariablet,thenwhentvariesbyanincrementt,rwillvarybyanincrementr.risavariableassociatedwithrbutitneedsnothaveeitherthesamemagnitudeofdirectionasr:编辑课件Astvaries,theendpointofthepositionvectorrwilltraceoutacurveinspace.Takingsasavariablemeasuringlengthalongthiscurve,thedifferentiationprocesscanbeperformedwithrespecttosthus:isaunitvectorinthedirectionofthetangenttothecurveisperpendiculartothetangent.Thedirectionofisthenormaltothecurve,andthetwovectorsdefinedasthetangentandnormaldefinewhatiscalledthe“osculatingplane”ofthecurve.编辑课件Temperatureisascalarquantitywhichcandependingeneraluponthreecoordinatesdefiningpositionandafourthindependentvariabletime.isa“partialderivative”.isthetemperaturegradientinthexdirectionandisavectorquantity.isascalarrateofchange.Partialdifferentiationofvectors编辑课件Adependentvariablesuchastemperature,havingtheseproperties,iscalleda“scalarpointfunction”andthesystemofvariablesisfrequentlycalleda“scalarfield”.Otherexamplesareconcentrationandpressure.Thereareotherdependentvariableswhicharevectorialinnature,andvarywithposition.Theseare“vectorpointfunctions”andtheyconstitute“vectorfield”.Examplesarevelocity,heatflowrate,andmasstransferrate.Scalarfieldandvectorfield编辑课件Hamilton’soperatorIthasbeenshownthatthethreepartialderivativesofthetemperaturewerevectorgradients.Ifthesethreevectorcomponentsareaddedtogether,thereresultsasinglevectorgradient:whichdefinestheoperatorfordeterminingthecompletevectorgradientofascalarpointfunction.Theoperatorispronounced“del”or“nabla”.ThevectorTisoftenwritten“gradT”forobviousreasons.canoperateuponanyscalarquantityandyieldavectorgradient.應用於scalar

的偏微编辑课件MoreabouttheHamilton’soperator...(vector)·(vector)ButTisthevectorequilvalentofthegeneralizedgradient

编辑课件Physicalmeaningof

T:Avariablepositionvectorrtodescribeanisothermalsurface:Sincedrliesontheisothermalplane…andThus,Tmustbeperpendiculartodr.Sincedrliesinanydirectionontheplane,Tmustbeperpendiculartothetangentplaneatr.ifA·B=0ThevectorAiszeroThevectorBiszero=90°drTTisavectorinthedirectionofthemostrapidchangeofT,anditsmagnitudeisequaltothisrateofchange.编辑课件Theoperatorisofvectorform,ascalarproductcanbeobtainedas:應用於vector

的偏微applicationTheequationofcontinuity:whereisthedensityanduisthevelocityvector.Output-input:thenetrateofmassflowfromunitvolumeAisthenetfluxofAperunitvolumeatthepointconsidered,countingvectorsintothevolumeasnegative,andvectorsoutofthevolumeaspositive.编辑课件AinAoutThefluxleavingtheoneendmustexceedthefluxenteringattheotherend.Thetubularelementis“divergent”inthedirectionofflow.Therefore,theoperatorisfrequentlycalledthe“divergence”:Divergenceofavector编辑课件Anotherformofthevectorproduct:isthe“curl”ofavector;Whatisitsphysicalmeaning?Assumeatwo-dimensionalfluidelementuvxyOABRegardedastheangularvelocityofOA,direction:kThus,theangularvelocityofOAis;similarily,theangularvelocityofOBis编辑课件Theangularvelocityuofthefluidelementistheaverageofthetwoangularvelocities:uvxyOABThisvalueiscalledthe“vorticity”ofthefluidelement,whichistwicetheangularvelocityofthefluidelement.Thisisthereasonwhyitiscalledthe“curl”operator.

编辑课件Wehavedealtwiththedifferentiationofvectors.Wearegoingtoreviewtheintegrationofvectors.编辑课件VectorintegrationLinearintegralsVectorareaandsurfaceintegralsVolumeintegrals编辑课件AnarbitrarypathofintegrationcanbespecifiedbydefiningavariablepositionvectorrsuchthatitsendpointsweepsoutthecurvebetweenPandQrPQdrAvectorA

canbeintegratedbetweentwofixedpointsalongthecurver:ScalarproductIftheintegrationdependsonPandQbutnotuponthepathr:ifA·B=0ThevectorAiszeroThevectorBiszero=90°编辑课件IfavectorfieldAcanbeexpressedasthegradientofascalarfield,thelineintegralofthevectorAbetweenanytwopointsPandQisindependentofthepathtaken.Ifisasingle-valuedfunction:and假如與從P到Q的路徑無關，則有兩個性質：Example:编辑课件Ifthevectorfieldisaforcefieldandaparticleatapointrexperiencesaforcef,thentheworkdoneinmovingtheparticleadistancerfromrisdefinedasthedisplacementtimesthecomponentofforceopposingthedisplacement:ThetotalworkdoneinmovingtheparticlefromPtoQisthesumoftheincrementsalongthepath.Astheincrementstendstozero:Whenthisworkdoneisindependentofthepath,theforcefieldis“conservative”.Suchaforcefieldcanberepresentedbythegradientofascalarfunction:Work,forceanddisplacementWhenascalarpointfunctionisusedtorepresentavectorfield,itiscalleda“potential”function:gravitationalpotentialfunction(potentialenergy)……………….gravitationalforcefieldelectricpotentialfunction………..electrostaticforcefieldmagneticpotentialfunction……….magneticforcefield编辑课件Surface:avectorbyreferecetoitsboundaryarea:themaximumprojectedareaoftheelementdirection:normaltothisplaneofprojection(right-handscrewrule)Thesurfaceintegralisthen:IfAisaforcefield,thesurfaceintegralgivesthetotalforaceactingonthesurface.IfAisthevelocityvector,thesurfaceintegralgivesthenetvolumetricflowacrossthesurface.编辑课件Volume:ascalarbyreferecetoitsboundaryABCBoththeelementsoflength(dr)andsurface(dS)arevectors,buttheelementofvolume(d)isascalarquantity.Thereisnomultiplicationforvolumeintegrals.Whataretherelationshipsbetweenthem?Stokes’theorem编辑课件SConsideringasurfaceShavingelementdSandcurveCdenotesthecurve:Stokes’Theorem（連接「線」和「面」的關係）

IfthereisavectorfieldA,thenthelineintegralofAtakenroundCisequaltothesurfaceintegralof×AtakenoverS:Two-dimensionalsystem编辑课件PQHowtomakealinetoasurface?PandQrepresentthesamepoint!你看到了一個「面」，你要如何去描述？從「線」著手從「面」著手编辑课件AinAoutThetubularelementis“divergent”inthedirectionofflow.ThenetrateofmassflowfromunitvolumeGauss’DivergenceTheorem（連接「面」和「體」的關係）

Wealsohave:ThesurfaceintegralofthevelocityvectorugivesthenetvolumetricflowacrossthesurfaceThemassflowrateofaclosedsurface(volume)编辑课件Gauss’DivergenceTheorem

（連接「面」和「體」的關係）

Stokes’Theorem（連接「線」和「面」的關係）

编辑课件UsefulequationsaboutHamilton’soperator...Aistobedifferentiatedvalidwhentheorderofdifferentiationisnotimportantinthesecondmixedderivative编辑课件CoordinatesotherthancartesianSphericalpolarcoordinates(r,,)Fig7.15theedgeoftheincrementelementisgeneralcurved.Ifa,b,careunitvectorsdefinedaspointP:

编辑课件ThegradientofascalarpointfunctionU:AssumingthatthevectorAcanberesolvedintocomponentsintermsofa,b,andc:编辑课件CoordinatesotherthancartesianCylindricalpolarcoordinates(r,,z)Fig7.17theedgeoftheincrementelementisgeneralcurved.Ifa,b,careunitvectorsdefinedaspointP:

编辑课件ThegradientofascalarpointfunctionU:AssumingthatthevectorAcanberesolvedintocomponentsintermsofa,b,andc:编辑课件Howcanweusevectorsinchemicalengineeringproblems?WhytheHamilton’soperatorisimportantforchemicalengineers?编辑课件Consideringthestudyof“fluidflow”,theheatingeffectduetofrictionandmasstransferareignored:Newtonianfluid :coefficientofviscosityremainsconstantIndependentvariables :x,y,zandtimeDependentvariables :u,v,w,pressure,density5dependentvariables5equations:(1)continuityequation(massbalance)(2)equationofstate(densityandpressure)(3)~(5)Newton’ssecondlawofmotiontoafluidelement(relatingexternalforces,pressureforce,viscousforcestotheaccelerationoffluidelement)Navier-StokesequationSolvetogether?编辑课件Stokes’Approximation(omittheinertiaterm,Re<<1)dimensionlessformdimensionlessgroupsdimensionlesstimedimensionlesspressurecoefficientReynoldsnumberincompressiblenotuseful,usuallyu,notp,isgivenvorticityanalogoustotheheatandmasstransferequation编辑课件TheidealfluidApproximation(omittheviscousandinertiaterm,Re>>)ifsteadystateandvorticity=0Bernoulli’sequation:(1)laminarflowissteady(2)imcompressible(3)inviscid(4)irrotationalincompressibleThevorticityofanyfluidelementremainsconstant.编辑课件ifafluidmotionstartsfromrest,thevorticityiszeroandflowisirrotationalReca