美赛A题O奖论文.pdf

美赛A题O奖论文.pdf

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2016MathematicalContestinModeling（MCM/ICM）SummarySheetSummaryAtraditionalbathtubcannotbereheatedbyitself,sousershavetoaddhotwaterfromtimetotime.Ourgoalistoestablishamodelofthetemperatureofbathwaterinspaceandtime.Thenweareexpectedtoproposeanoptimalstrategyforuserstokeepthetemperatureevenandclosetoinitialtemperatureanddecreasewaterconsumption.Tosimplifymodelingprocess,wefirstlyassumethereisnopersoninthebathtub.Weregardthewholebathtubasathermodynamicsystemandintroduceheattransferformulas.Weestablishtwosub-models:

addingwaterconstantlyanddiscontinuously.Asfortheformersub-model,wedefinethemeantemperatureofbathwater.IntroducingNewtoncoolingformula,wedeterminetheheattransfercapacity.Afterderivingthevalueofparameters,wededuceformulastoderiveresultsandsimulatethechangeoftemperaturefieldviaCFD.Asforthesecondsub-model,wedefineaniterationconsistingoftwoprocess:

heatingandstandby.Accordingtoenergyconservationlaw,weobtaintherelationshipoftimeandtotalheatdissipatingcapacity.Thenwedeterminethemassflowandthetimeofaddinghotwater.WealsouseCFDtosimulatethetemperaturefieldinsecondsub-model.Inconsiderationofevaporation,wecorrecttheresultsofsub-modelsreferringtosomescientistsstudies.Wedefinetwoevaluationcriteriaandcomparethetwosub-models.Addingwaterconstantlyisfoundtokeepthetemperatureofbathwaterevenandavoidwastingtoomuchwater,soitisrecommendedbyus.Thenwedeterminetheinfluenceofsomefactors:

radiationheattransfer,theshapeandvolumeofthetub,theshape/volume/temperature/motionsoftheperson,thebubblesmadefrombubblebathadditives.Wefocusontheinfluenceofthosefactorstoheattransferandthenconductsensitivityanalysis.Theresultsindicatesmallerbathtubwithlesssurfacearea,lighterpersonalmass,lessmotionsandmorebubbleswilldecreaseheattransferandsavewater.Basedonourmodelanalysisandconclusions,weproposetheoptimalstrategyfortheuserinabathtubandexplainthereasonofuneventemperaturethroughoutthebathtub.Inaddition,wemakeimprovementforapplyingourmodelinreallife.Keywords:

HeattransferThermodynamicsystemCFDEnergyconservationForofficeuseonlyT1T2T3T4ForofficeuseonlyF1F2F3F4TeamControlNumber44398ProblemChosenATeam#44398Page2of51EnjoyaCozyandGreenBathConContentstents1Introduction.41.1Background.41.2LiteratureReview.41.3RestatementoftheProblem.42AssumptionsandJustification.63Notations.74ModelOverview.75Sub-modelI:

AddingWaterContinuously.85.1ModelEstablishment.95.1.1ControlEquationsandBoundaryConditions.95.1.2DefinitionoftheMeanTemperature.115.1.3DeterminationofHeatTransferCapacity.115.2Results.135.2.1DeterminationofParameters.135.2.2CalculatingResults.145.2.3SimulatingResults.156Sub-modelII:

AddingWaterDiscontinuously.186.1HeatingModel.186.1.1ControlEquationsandBoundaryConditions.186.1.2DeterminationofInflowTimeandAmount.196.2StandbyModel.206.2.1ProcessAnalysis.206.2.2CalculationofParameters.206.3Results.216.3.1DeterminationofParameters.216.3.2CalculatingResults.23Team#44398Page3of516.3.3SimulatingResults.236.4Conclusion.277CorrectionandContrastofSub-Models.277.1CorrectionwithEvaporationHeatTransfer.277.1.1CorrectionPrinciple.277.1.2CorrectionResults.287.2ContrastofTwoSub-Models.307.2.1EvaluationCriteria.307.2.2DeterminationofWaterConsumption.307.2.3Conclusion.318ModelAnalysisandSensitivityAnalysis.318.1TheInfluenceofDifferentBathtubs.328.1.1DifferentVolumesofBathtubs.328.1.2DifferentShapesofBathtubs.348.2TheInfluenceofPersoninBathtub.368.2.1WhenthePersonRemainsStaticinaBathtub.368.2.2WhenthePersonMovesinaBathtub.378.2.3ResultsAnalysisandSensitivityAnalysis.388.3TheInfluenceofBubbleBathAdditives.428.4TheInfluenceofRadiationHeatTransfer.448.5Conclusion.459FurtherDiscussion.459.1DifferentDistributionofInflowFaucets.459.2ModelApplication.4610StrengthandWeakness.4710.1Strength.4710.2Weakness.47Report.49Reference.50Team#44398Page4of511Introduction1.1BackgroundBathinginatubisaperfectchoiceforthosewhohavebeenwornoutafteralongdaysworking.Atraditionalbathtubisasimplywatercontainmentvesselwithoutasecondaryheatingsystemorcirculatingjets.Thusthetemperatureofwaterinbathtubdeclinesnoticeablyastimegoesby,whichwillinfluenttheexperienceofbathing.Asaresult,thebathingpersonneedstoaddaconstanttrickleofhotwaterfromafaucettoreheatthebathingwater.Thiswayofbathingwillresultinwasteofwaterbecausewhenthecapacityofthebathtubisreached,excesswateroverflowsthetub.Anoptimalbathingstrategyisrequiredforthepersoninabathtubtogetcomfortablebathingexperiencewhilereducingthewasteofwater.1.2LiteratureReviewKoreanphysicistGi-BeumKimanalyzedheatlossthroughfreesurfaceofwatercontainedinbathtubduetoconductionandevaporation1.Hederivedarelationalequationbasedonthebasictheoryofheattransfertoevaluatetheperformanceofbathtubes.Themajorheatlosswasfoundtobeduetoevaporation.Moreover,hefoundoutthatthespeedofheatlossdependsmoreonthehumidityofthebathroomthanthetemperatureofwatercontainedinthebathtub.So,itisbesttomaintainthetemperatureofbathtubwatertobebetween41to45andthehumidityofbathroomtobe95%.Whenitcomestoconvectiveheattransferinbathtub,manystudiescanbereferredto.Newtonslawofcoolingstatesthattherateofheatlossofabodyisproportionaltothedifferenceintemperaturesbetweenthebodyanditssurroundingswhileundertheeffectsofabreeze2.Claude-LouisNavierandGeorgeGabrielStokesdescribedthemotionofviscousfluidsubstanceswiththeNavierStokesequations.Thoseequationsmaybeusedtomodeltheweather,oceancurrents,waterflowinapipeandairflowaroundawing3.Inaddition,somenumericalsimulationsoftwareareappliedinsolvingandanalyzingproblemsthatinvolvefluidflows.Forexample,ComputationalFluidDynamics（CFD）isacommononeusedtoperformthecalculationsrequiredtosimulatetheinteractionofliquidsandgaseswithsurfacesdefinedbyboundaryconditions4.1.3RestatementoftheProblemWearerequiredtoestablishamodeltodeterminethechangeofwaterTeam#44398Page5of51temperatureinspaceandtime.Thenweareexpectedtoproposethebeststrategyforthepersoninthebathtubtokeepthewatertemperatureclosetoinitialtemperatureandeventhroughoutthetub.Reductionofwasteofwaterisalsoneeded.Inaddition,wehavetoconsidertheimpactofdifferentconditionsonourmodel,suchasdifferentshapesandvolumesofthebathtub,etc.Inordertosolvethoseproblems,wewillproceedasfollows:

Statingassumptions.Bystatingourassumptions,wewillnarrowthefocusofourapproachtowardstheproblemsandprovidesomeinsightintobathtubwatertemperatureissues.Makingnotations.Wewillgivesomenotationswhichareimportantforustoclarifyourmodels.Presentingourmodel.Inordertoinvestigatetheproblemdeeper,wedivideourmodelintotwosub-models.Oneisasteadyconvectionheattransfersub-modelinwhichhotwaterisaddedconstantly.Theotheroneisanunsteadyconvectionheattransfersub-modelwherehotwaterisaddeddiscontinuously.Definingevaluationcriteriaandcomparingsub-models.Wedefinetwomaincriteriatoevaluateourmodel:

themeantemperatureofbathwaterandtheamountofinflowwater.Analysisofinfluencingfactors.Intermoftheimpactofdifferentfactorsonourmodel,wetakethoseintoconsideration:

theshapeandvolumeofthetub,theshape/volume/temperatureofthepersoninthebathtub,themotionsmadebythepersoninthebathtubandaddingabubblebathadditiveinitially.Modeltestingandsensitivityanalysis.Withthecriteriadefinedbefore,weevaluatethereliabilityofourmodelanddothesensitivityanalysis.Furtherdiscussion.Wediscussaboutdifferentwaystoarrangeinflowfaucets.Thenweimproveourmodeltoapplytheminreality.Evaluatingthemodel.Wediscussaboutthestrengthsandweaknessesofourmodel.Team#44398Page6of512AssumptionsandJustificationTosimplifytheproblemandmakeitconvenientforustosimulatereal-lifeconditions,wemakethefollowingbasicassumptions,eachofwhichisproperlyjustified.ThebathwaterisincompressibleNon-Newtonianfluid.TheincompressibleNon-NewtonianfluidisthebasisofNavierStokesequationswhichareintroducedtosimulatetheflowofbathwater.Allthephysicalpropertiesofbathwater,bathtubandairareassumedtobestable.Thechangeofthosepropertieslikespecificheat,thermalconductivityanddensityisrathersmallaccordingtosomestudies5.Itiscomplicatedandunnecessarytoconsidertheselittlechangesoweignorethem.Thereisnointernalheatsourceinthesystemconsistingofbathtub,hotwaterandair.Beforethepersonliesinthebathtub,nointernalheatsourceexistexceptthesystemcomponents.Thecircumstancewherethepersonisinthebathtubwillbeinvestigatedinourlaterdiscussion.Weignoreradiativethermalexchange.AccordingtoStefan-Boltzmannslaw,theradiativethermalexchangecanbeignoredwhenthetemperatureislow.Refertoindustrialstandard6,thetemperatureinbathroomislowerthan100,soitisreasonableforustomakethisassumption.Thetemperatureoftheaddinghotwaterfromthefaucetisstable.Thishypothesiscanbeeasilyachievedinrealityandwillsimplifyourprocessofsolvingtheproblem.Team#44398Page7of513NotationsTable1NotationsSymbolsDefinitionUnithConvectionheat