最新年美赛A题H奖58265390文档格式.docx

最新年美赛A题H奖58265390文档格式.docx

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1.2Background3

1.3ProblemDescription3

2.Modelforheatdistribution3

2.1Problemanalysis3

2.2Assumptions4

2.3Definitions4

2.4Themodel4

3Resultsofheatdistribution7

3.1Basicresults7

3.2Analysis9

3.3Analysisofthetransitionshape—roundedrectangular9

4Modeltoselectthebestshape11

4.1Assumptions11

4.2Definitions11

4.3Themodel12

5ComparisionandDegreeoffitting19

6Sensitivity20

7Strengths/weaknesses21

8Conclusions21

9AdvertisementfornewBrownieMagazine23

10References24

1Introduction

1.1Browniepan

TheBrowniePanisusedtomakeBrownieswhichareakindofpopularcakesinAmerica.Itusuallyhasmanylatticesinitandismadeofmetalorothermaterialstoconductheatwell.Itistrivially9×

9inchor9×

13inchinsize.OneexampleoftheconcreteshapeofBrowniepanisshowninFigure1

Figure1theshapeofBrowniePan（source:

GoogleImage）

1.2Background

BrowniesaredeliciousbuttheBrowniePanhasafetaldrawback.Whenbakinginarectangularpan,thefoodcaneasilygetovercookedinthe4corners,whichisveryannoyingforthegreedygourmets.Inaroundpan,theheatisevenlydistributedovertheentireouteredgebutisnotefficientwithrespecttousinginthespaceinanoven,whichmostcakebakerswouldnotliketosee.Soourgoalistoaddressthisproblem.

1.3ProblemDescription

Firstly,weareaskedtodevelopamodeltoshowthedistributionofheatacrosstheouteredgeofapanfordifferentshapes,fromrectangulartocircularincludingthetransitionshapes;

thenwewillbuildanothermodeltoselectthebestshapeofthepanfollowingtheconditionoftheoptimizationofcombinationsofmaximalnumberofpansintheovenandmaximalevendistributionofheatforthepan.

2.Modelforheatdistribution

2.1Problemanalysis

Hereweuseafinitedifferencemodeltoillustratethedistributionofheat,andithasbeenextensivelyusedinmodelingforitscharacteristicabilitytohandleirregulargeometriesandboundaryconditions,spatialandtemporalpropertiesvariations.Inliterature1,sampleswitharectangulargeometricformaredifficulttoheatuniformly,particularlyatthecornersandedges.Theythinkmicrowaveradiationintheovencanbecrudelythoughtofasimpingingonthesamplefromall,whichwegenerallyacknowledge.Buttheyemphasizetherotation.

Generally,whenbakingintheoven,thecakesabsorbheatbythreeways:

thermalradiationofthepipesintheoven,heatconductionofthepan,andairconvectionintheoven.Consideringthattheinfluenceofconvectionissmall,weassumeitnegligible.Soweonlytakethermalradiationandconductionintoaccount.Theheatistransferredfromtheoutsidetotheinsidewhilewaterinthecakeisonthecontrary.Thetemperatureoutsideincreasemorerapidlythanthatinside.Andthecontactareabetweenthepanandtheoutsidecakeislargerthanthatbetweenthepanandtheinsidecakes,whichillustratewhycakesinthecornergetovercookedeasily.

2.2Assumptions

●Wetakethepanandcakesasblackbody,sotheabsorptionofheatineachareaunitandtimeunitisthesame,whichdrasticallysimplifiesourcalculation.

●Weassumetheairconvectionnegligible,consideringitscomplexityandthesmallinfluenceonthetemperatureincrease.

●Weneglecttheevaporationofwaterinsidethecake,whichmayimpedetheincreaseoftemperatureofcakes.

●Weignorethethicknessofcakesandthepan,sothemodelwebuildistwo-dimensional.

2.3Definitions

Φ:

heatflowsintothenode

Q:

theheattakeninbycakesorpansfromtheheatpipes

:

energyincreaseofeachcakeunit

energyincreaseofthepanunit

temperatureatmomentiandpoint（m,n）

C1:

thespecificheatcapacityofthecake

C2:

thespecificheatcapacityofthepan

temperatureofthepanatmomenti

T1:

temperatureintheoven,whichweassumeisaconstant

2.4Themodel

Hereweusefinite-differencemethodtoderivetherelationshipoftemperaturesattimei-1andtimeiatdifferentplaceandtherelationshipoftemperaturesbetweenthepanandthecake.

Firstwedivideacakeintosmallunits,whichcanbeexpressedbyametric.Inthefollowingsection,wewilldiscussthecakeunitindifferentplacesofthepan.

Step1；

temperaturesofcakesinterior

Figure2heatflow

Accordingtoenergyconversationprinciple,wecanget

（2.4.1）

ConsideringFourierLawand△x=△y,weget

（2.4.2）

AccordingtoStefan-BoltzmanLaw,

（2.4.3）

WhereAistheareacontacting,cistheheatconductance.σistheStefan-Boltzmannconstant,andequals5.73×

108Jm-2s-1k-4.

（2.4.4）

Substituting（2.4.2）-（2.4.4）into（2.4.1）,weget

Thisequationdemonstratestherelationshipoftemperatureatmomentiandmomenti-1aswellastherelationshipoftemperatureat（m,n）anditssurroundingpoints.

Step2:

temperatureofthecakeouterandthepan

●Forthe4corners

Figure3therelativepositionofthecakeandthepaninthefirstcorner

Becausethecontactingareaistwotimes,weget

●Foreveryedge

Figure4therelativepositionofthecakeandthepanattheedge

Similarly,wederive

Nowthatwehavederivedtheexpressoftemperaturesofcakesbothtemporallyandspatially,wecanuseiterationtogetthecurveoftemperaturewiththevariables,timeandlocation.

3Resultsofheatdistribution

3.1Basicresults

●Rectangular

Preliminarily,wefocusononecorneronly.Afterrunningtheprogramme,weobtainthefollowingfigure.

Figure5heatdistributionatonecorner

Figure5demonstratesthetemperatureatthecornerishigherthanitssurroundingpoints,that’swhyfoodatcornersgetovercookedeasily.

Thenweiterateglobally,andgetFigure6.

Figure6heatdistributionintherectangularpan

Figure6canintuitivelyillustratesthetemperatureatcornersisthehighest,andtemperatureontheedgeislesshigherthanthatatcorners,butismuchhigherthanthatatinteriorpoints,whichsuccessfullyexplainstheproblem“productsgetovercookedatthecornersbuttoalesserextentontheedge”.

Afterdrawingtheheatdistributionintwodimensions,wesamplesomepointsfromtheinsidetotheoutsideinarectangularandobtaintherelationshipbetweentemperatureanditerationtimes,whichisshowninFigure7

Figure7

FromFigure7,thetemperaturesgoupwithtimegoingandthenkeepnearlyparalleltothex-axis.Ontheotherhand,temperatureatthecenterascendstheslowest,thenedgeandcorner,whichmeansgivencookingtime,foodatthecenterofthepaniscookedjustwellwhilefoodatthecornerofthepanhasalreadygetovercooked,butalesserextenttotheedge.

●Round

Weuseourmodeltoanalyzetheheatdistributioninaround,justadaptingtherectangularunitsintosmallannuluses,byrunningourprogramme,wegetthefollowingfigure.

Figure8theheatdistributioninthecirclepan

Figure8showsheatdistributionincircleareaiseven,theproductsattheedgearecookedtothesameextentapproximately.

3.2Analysis

Finally,wedrawtheisothermalcurveofthepan.

Figure9theisothermalcurveoftherectangularpan

Figure9demenstratestheisothermallinesarealmostconcentriccirclesinthecenterofthepanandbecomeroundedrectanglesouter,whichprovidestheorysupportforfollowinganalysis..

●Circular

Figure10theisothermalcurveofcircular

Theisothermalcurvesofthecircularareseriesofconcentriccircles,demonstratingthattheheatisevendistributed.

3.3Analysisofthetransitionshape—roundedrectangular

Fromtheaboveanalysis,wefindthattheisothermalcurvearenearlyroundedrectangularsintherectangularpan,soweperspectivethetransitionshapebetweenrectangularandcircularisroundedrectangular,consideringtheefficiencyofusingspaceoftheovenandtheevenheatdistribution.Inthefollowingsection,wewillanalyzetheheatdistributioninroundedrectangularpanusingfiniteelementapproach.

Duringthecookingprocess,thetemperaturegoesupgradually.Butatacertainmoment,thetemperaturecanbeassumedaconstant.SotheboundaryconditionyieldsDirichletboundary