2016年美国大学生数学建模大赛A题获奖论文A-Hot-Bath

SummaryOurexperienceofeverydaylifetellsusthat,bathingqualityisrelevanttothetemperatureofthewater.Hencetheoptimalityofthewater-addingstrategyinbathingprocesshasbecomeanissue.Inthisarticle,weestablishtwomodels.Oneiswatertemperaturevariationanddistributionmodelandtheotheroneisfindingbeststrategymodel.Weputforwardsomeacceptablehypothesistosimplifythemodel.What’smore,weclearthemeaningoftheword“noticeably”.Thefirstparthasfivesections:air’sheatradiation,bathtubwall’sheatradiation,personin,hotwaterin,bubbleexisted.Wediscusssomefactorsthataffectwatertemperature,suchastheshapeandthevolumeofthebathtubandperson,especiallythemotionsmadebythepersoninthebathtubbecausethetemperatureinthebathtubhasagreatconnectionwithperson.Finally,wegetthewatertemperaturevariationanddistributionmodel.Thesecondpartaimstofindingthebestwaterstrategy.Differentstrategiesareconsideredforaddinghotwaterandtheamountofthehotwater.Thekeyofstrategiesishowtochoosethemomentofopeningthefaucetandthelastingtime.Bydevelopinganoptimizationmodelofthestrategy,wemanagetodetermineastrategyfordifferentusersandextendthemodelwithafinitevolumemethod.Undertheconditionofthesamebathingtime,westudywhat’stheidealtimetoaddthehotwatertomakeittotheinitialtemperature.ForofficeuseonlyT1________________T2________________T3________________T4________________TeamControlNumber46731ProblemChosenAForofficeuseonlyF1________________F2________________F3________________F4________________2016MCM/ICMSummarySheet(Yourteam'ssummaryshouldbeincludedasthefirstpageofyourelectronicsubmission.)Typeasummaryofyourresultsonthispage.Donotincludethenameofyourschool,advisor,orteammembersonthispage.Team#46731Page2of19Wetransformtheweightp.Withtherestrictconditionofthetemperatureinthebathandtheamountofthehotwater,thequestioncanbetransformedtoanonlinearoptimization.WegettheconclusionandoptimalresultsthroughthePDE(PartialDifferentialEquations)toolboxofMATLAB.Andtheresultshowsaclearsuperiorityforchoosingasuitablestrategyfordifferentusers.Keywords:bathtub;heat-conduction;temperaturevariationanddistribution;waterstrategy;finitevolumemethod;Team#46731Page3of19ContentsSummary................................................................................................................1I.Introduction......................................................................................................4II.Analysis........................................................................................................4III.Assumptions.................................................................................................6IV.Notation........................................................................................................7V.Models..........................................................................................................8PartIWaterTemperatureVariationandDistributionModel.........................8Air’sHeatRadiation..........................................................................8BathtubWall’sHeatRadiation...........................................................9PersonIn............................................................................................9HotWaterIn.....................................................................................10BubbleExisted.................................................................................11WaterTemperatureVariationandDistributionModel.....................11PartIIFindingthebeststrategy...................................................................12VI.Results........................................................................................................12PartI.............................................................................................................12ResultsofAir’sHeatRadiationModel............................................12ResultsofBathtubWall’sHeatRadiation........................................13Resultsof“PersonIn”......................................................................13Resultsof“HotWaterIn”................................................................13Resultsofbubbleexistedmodel......................................................15ResultsofWaterTemperatureVariationandDistributionModel....15PartII...........................................................................................................16VII.StrengthsandWeaknesses.......................................................................17VIII.Explanationforusers..............................................................................18IX.References......................................................................................................19Team#46731Page4of19I.IntroductionAhotwaterbathcanrelaxpeople'sbodyanditisoneofthemostpopularwaystorelaxeversince.However,peopleareoftentroubledbythewaterbecauseitisgettingcolderandcolder.Inthiscase,itiscrucialtostudythewatertemperaturechangeswithspaceandtime,andtoprovidepeoplewiththebestbathstrategy.Theinteractionandsysteminthebatisrathercompl