(5.4)--Chapter 4 Numerical solution of传热学传热学传热学.ppt
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1、Chapter4NumericalsolutionofheatconductionproblemsnThebasicprincipleofnumericalsolutionofheatconductionnMethodforestablishingdiscreteequationsofinteriornodesnTheestablishmentandsolutionofdiscreteequationsofsurfacenodesnNumerical solution of unsteady heatconductionproblem1.Threebasicmethodsforsolvingt
2、hermalproblems:2.Thebasicsolutionprocessofthethreemethods(1)theoreticalanalysismethod;(2)numericalcalculation;(3)experimentmethod(1)Theso-calledtheoreticalanalysismethodistodirectlyintegrate the finite difference equation under a givendefiniteconditionbasedontheoreticalanalysis,sothattheobtained sol
3、ution is called an analytical solution,or atheoreticalsolution;Solutionofheatconductionproblem(3)Experimentalmethod:Undertheguidanceofthebasictheory of heat transfer,obtain the required quantity byexperimentingonheattransferprocessoftheresearchobject.(2)Fornumericalmethod,thefieldofthephysicalquanti
4、tythatiscontinuousintimeandspaceisreplacedbyasetofvaluesatafinitenumberofdiscretepoints,andthealgebraicequationsaboutthesevaluesbyacertainmethodaresolvedtoobtainthevaluesofthephysicalquantityatthesediscretepointsandcallitnumericalsolution.(1)Theoreticalanalysismethoda.Anaccuratesolutiontothestudiedp
5、roblemcanprovideabasisforcomparisonbetweenexperimentsandnumericalcalculations;b.Analyticalsolutionsareuniversalandtheeffectsofvarioussituationsareclearlyvisible;c.Thelimitationislargeandcannotbesolvedforcomplexproblems.3.Characteristicsofthethreemethods(2)Numericalmethod:Itlargelyovercomestheshortco
6、mingsoftheoreticalanalysismethodandhasstrongadaptability,especiallyforcomplexproblems.Ithaslowercostthanexperimentalmethod.(3)Experimentalmethod:basicresearchmethodofheattransfera.Badadaptability;b.ExpensiveFinite-differenceFinite-elementBoundary-element4.1Basicideaofnumericalsolutionofheatconductio
7、nproblem1.BasicideaThebasicideaofnumericallysolvingphysicalproblemscanbesummarizedas:Replacethecontinuousfieldofphysicalquantitiesinthetimeandspacecoordinatesystem(suchasthetemperaturefieldoftheheat-conductingobject)withasetofvaluesatafinitenumberofdiscretepoints.Obtainthevalueofphysicalquantityatdi
8、scretepointsbysolvingthealgebraicequationsaboutthesevaluesestablishedbyacertainmethod.2.Stepsfornumericalsolution(1)EstablishgoverningequationsandsetconditionsGoverningequations:Definiteconditions:Heattransferproblemfortwo-dimensional,steady-state,withoutinternalheatsource,andconstantphysicalpropert
9、iesinrectangulardomain(2)RegionaldiscretizationAsshownStepsizeThe intersections of gridlinesareknotpointElement:The small arearepresented by nodes iscalled the element,alsocalledthecontrolvolume.(3)Establishthealgebraicequationsforphysicalquantitiesofnodes(discreteequations)Algebraicequationfornode(
10、m,n),when(4)EstablishaniterativeinitialfieldTherearetwomaintypesofmethodsforsolvingalgebraicequations:directsolutionanditerativesolution.Whenusingiterativesolution,itisnecessarytopresupposeasolutiontotherequestedphysicalfield,calledinitialfield,andtheinitialfieldiscontinuouslyimprovedduringthesoluti
11、onprocess.(5)SolvethealgebraicequationsProblemsinsolving:1)linear;2)nonlinear;3)convergence,etc.Linearequations:coefficientsinalgebraicequationsdonotchangeduringtheentiresolution;Nonlinearequations:coefficientsinalgebraicequationsarecontinuouslyupdatedthroughoutthesolution.Determinewhetheritconverge
12、s:referstotheconvergenceofalgebraicequationsusingiterativemethod,thatis,whetherthedeviationofthesolutioncalculatedbythisiterationandthepreviousiterationislessthantheallowablevalue.(6)AnalysisofthesolutionHeatrate,thermalstress,thermaldeformationFlowdiagramImprovetheinitialfieldNoIf,,findConvergenceA
13、nalysisofsolutionsYesEstablishcontrolequationsandsetconditionsDeterminenodes(areadiscretization)EstablishalgebraicequationsforphysicalquantitiesofnodesSetiterativeinitialvalueoftemperaturefieldSolvealgebraicequations1.RegiondiscretizationThedivisionoftheareadependsonthegeometricconvenienceandtherequ
14、iredcalculationaccuracy.Nodes,Griddense,calculationaccuracy,Longcalculationtime4.2Methodforestablishingdiscreteequationsofinternalnodesiscallednodes,meansunit(temperature,properties).2.Commonmethodsforbuildingdiscreteequations(1)Taylorseriesexpansion(2)Controlvolumebalancemethod、(alsoknownasthermalb
15、alancemethod)(1)TaylorseriesexpansionAccordingtoTaylorseriesexpansion,representthetemperatureofnode(m+1,n),tm+1,n,withthetemperatureofnode(m,n),tm,n:thenrepresentthetemperatureofnode(m-1,n),tm-1,n,withthetemperatureofnode(m,n),tm,n:AddthelasttwoformulasTruncationerrorGetsecond-ordercenterdifference:
16、Nodeequationis:=If,thenForsteady-stateheatconductionproblemoftwo-dimensional,heat conduction finite difference equation inCartesiancoordinatesis:(2)ThermalbalancemethodBasic idea:apply energy conservation to each finite-size controlvolumetoobtainalgebraicequationsoftemperaturefield.Itderivesfrombasi
17、cphysicalphenomenaandbasiclawswithoutestablishinggoverning equations in advance,and solve directly according toenergyconservationandFourierslawofheatconduction.Energyconservation:Totalheatrateintocontrolvolume+heatgenerationincontrolvolume=Total heat rate out control volume+Internal energy increment
18、 ofcontrolvolumeThatis:i.e.Totalheatrateintocontrolvolumefromalldirections+Heatgeneratedbyinternalheatsourceofcontrolvolume=InternalenergyincrementofcontrolvolumeNote:Above formula applies to both interior and surface nodes.Forsteady-state,noheatgenerationcondition:Totalheatrateintocontrolvolumefrom
19、alldirections=0Interiornodes:(m,n)oyx(m-1,n)(m+1,n)(m,n-1)x x y y(m,n+1)Fortwo-dimensional,steady-state,withinternalheatgeneration:Evidently:cantbeobtainedbeforetemperaturefieldis calculated.Therefore,temperature distribution betweenadjacentnodesmustbeassumedandheretobeapiecewiselineardistribution.(
20、m,n)(m-1,n)(m+1,n)tm,ntm-1,ntm+1,nInternalheatsource:whenThen:Whenthereisnointernalheatsource,itbecomes:Importantnote:Thecoefficientinfrontoftemperatureofnodetobedeterminedmustbeequaltothesumofcoefficientsinfrontoftemperatureofallotheradjacentnodes.Thisconclusionalsoappliestosurfacenodes.However,the
21、coefficientinfrontofheatrate(orheatflux)isnotincludedhere.4.3EstablishmentofdiscreteequationsofsurfacenodesandsolutionofalgebraicequationsForheatconductionproblemwithfirstboundaryconditions,processisrelativelysimple.Becausesurfacetemperatureisknownanditcanbeaddedtodiscreteequationsofinteriornodesinf
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