2016年美赛A题西南交通大学获奖论文

ForofficeuseonlyT1________________T2________________T3________________T4________________TeamControlNumber46830ProblemChosenAForofficeuseonlyF1________________F2________________F3________________F4________________2016MCM/ICMSummarySheetKeepItWarmSummaryDoyouusuallygetannoyedwiththecoolingofwaterduringarelaxingbath?Inthispaper,severalmodelsareproposedtosolvetheproblem.Oureffortsinbuildingthesemodelsfollowedthepathbelow.Firstly,auniversalthermodynamicmodel(UTM)ofbathwaterisestablishedbyconstructingtwoinitialsub-equationsnamedtheHeatDifferentialEquation(HDE)andtheTemperatureDistribution(TD)respectively.BasedonFourier'slaw,theStefan-Boltzmannlawandthefirstlawofthermodynamics,HDEisestablished,whichalsotakesheatconduction,heatradiationandevaporationofwaterintoaccount.Moreover,wegetthenumericalsolutionofTDbyusingthePDEtoolofMATLAB,andfititstemperaturedistributionsurfacebytheplaneequation,whosegoodnessoffitis0.931.Thefittingequationisregardedastheapproximateanalyticalsolution.Secondly,takingconvectioncausedbyhumanmovementandcomfortofwatertemperatureintoaccount,wedevelopedanoptimizationcontrolmodelforthebathstrategy(OCM)basedonUTM,whichaimsatthesmallestwaterinjection,thebesttemperaturestability&thehighestcomfortlevel.Weusethemodel(OCM)tooptimizethetwofollowingstrategies：i.Injectinghotwaterconstantlyatacertainflow;ii.Injectinghotwaterwhentemperaturedeclinestomin1Tandstoppinginjectingwhentemperatureincreasetomin2T.InstrategyI,theoptimalconditionsarethattheflowis10.007Lsandthetemperatureofwaterinjectionis311.1K.Theoptimizationprovesefficientafterextensivediscussion;InstrategyII,theupperandlowercriticaltemperatureandthetemperatureofwaterinjectionare311.7K，311.7K&313.1Krespectively.Itmeansthatthehotwaterinjectionisacontinuousprocess.Thirdly,wemodifyUTMwiththeactualsituationofbubblewatertoanalyzetheinfluenceofbubblewatertowardstheoptimization,concludingthattheheatdissipatingpowerofbubblewateris45.85%ofthatofnormalhotwaterat312K.ByOCM,wegettheoptimalconditionsastheflowis10.004Lsandthetemperatureofwaterinjectionis311.9K,whichshowsthattheefficiencyofbubblewaterexceedgreatlythatofnormalhotwater.Finally,wemadeerroranalysisofthefactorslikeexternalenvironmentetc.anddidstabilityanalysisofthegeometricparametersofthebathtub&theparametersofhumanbody.Thesensitivitiesofthefactorsprovethemodelstobefeasibleandapplicable.Team#46830Page1of45Contents1Introduction..................................................................................................11.1RestatementoftheProblem.....................................................................11.2OurApproach...........................................................................................12AssumptionswithJustifications..................................................................22.1AbouttheBathroomandWater................................................................32.2AbouttheHumanBody...........................................................................33Notations......................................................................................................44TheModel....................................................................................................54.1NormalThermodynamicModelofWaterTemperature...........................64.1.1Thedifferentialequationofwaterheatinthetub..........................64.1.2Thermodynamicanalysisofbathprocess...................................104.1.3Temperaturefielddistributionofthewaterinthetub.................114.1.4Theuniversalthermodynamicmodelofwater...........................164.2Forcedconvectionmodel.......................................................................174.2.1Theintroductionofforcedconvectionsource............................174.2.2Effectofforcedconvectiononthetemperature..........................184.3Thehumanbodycomfortmodel............................................................204.3.1Establishhumanbodycomfortmodel........................................204.3.2Solutionofhumanbodycomfortmodel.....................................214.4Optimizedcontrolmodelofbathingstrategy........................................224.4.1StrategyI.....................................................................................234.4.2StrategyII....................................................................................254.4.3SolutionoftheoptimalcontrolmodelsofstrategyI&II...........274.5Bubblebath............................................................................................314.5.1Theeffectofbubblewater..........................................................314.5.2Effectofbubblewateronoptimalcontrolstrategy.....................325TestingtheModel.......................................................................................345.1ErrorAnalysis........................................................................................345.1.1Variousfactorsofexternalenvironment.....................................345.1.2Variousfactorsofwater....................................