美国大学生数学建模竞赛2013 获奖论文

美国大学生数学建模竞赛

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TheUltimateBrowniePan

JiWang

FengHuang

CongliFan

UniversityofElectronicScienceandTechnologyofChina

Advisor:YongZhang

Summary

Makingdeliciousbrownieshasacloserelationshipwiththeshapeofbakingpans,howevertheproblemdepressingussomuchisthatwhenbakinginrectangularpans,theheatconcentratesonthecornercausingthefoodgetovercooked.Meanwhile,bakinginroundpansisnotefficientwithrespecttousingthespaceinarectangularoven.Nowtherefore,wedevelopoptimalmodelstoselectthebesttypeofpans.

Maximizeevendistributionofheatforthepans,itisnecessaryforustofigureouttheheatdistributionfordifferentshapes’pansprimarily.Basedonthesearchingresults,weuseTwo-DimensionalHeatConductionEquationtodescribethedistributionofheatacrosstheouteredgeofpans.Moreover,wesimulatetheheatdistributionofrectangular,roundandpolygonalpansbyusingthePEDtoolboxinMATLAB.

Forthecaseofthemaximumnumberofpansthatcanfitintheoven,weadoptRectangularPackingAlgorithmtosimplifytheproblem.Fromthiswedemonstrateageneralmethodforchoosingtheshapeofpanstomaximizethespaceutilization.Foranovenwithdeterminedradioofwidthtolength,itcancontainmorerectangularpansthanroundandpolygonalpans.

Inaddition,wesetupanindex definedasthestandarddeviationofthetemperatureinthedomainofpanstorepresenttheuniformityofheatdistribution.Andwedrawaconclusionthattheheatdistributionofroundpansisevenerthanregularpolygonalpans.Meanwhile,forrectangularpans,whentheratioofthelengthtothewidthisnearby1:1,itisaworsechoicethanroundpans.Butwhentheratioislesserthanthe1.0:1.8,therectangularpanscanbeabetterchoice.

Whentwofactorsabovearetakenintoaccount,wedevelopanoptimalmodeltodeterminethefinalbestbakingpan.Finally,wecometoaconclusionthattheoptimalshapeofpansvarieswithvaluesofW/Landp.

Finally,wecommentthestrengthsandweaknessesonourmodels.

KeywordsHeatConductionEquation,PEDtoolbox,RectangularPacking

美国大学生数学建模竞赛

Introduction

Whenbakinginarectangularpan,heatisconcentratedinthe4cornersandtheproductgetsovercookedatthecorners(andtoalesserextentattheedges).Inaroundpan,theheatisdistributedevenlyovertheentireouteredgeandtheproductisnotovercookedattheedges.So,heatcannotcirculateevenlyinsidethepan,leadingtotheboundarytemperaturehigherthaninner.

However,sincemostovensarerectangularinshape,usingroundpansisnotefficientwithrespecttousingthespaceinanoven.Thus,searchingforthemostsuitableshapeofthepansintheovenisverybeneficial.

Toexplorethemostsuitableshape,weshouldtaketwoaspectsintoconsideration:

1.Maximizenumberofpansthatcanfitintheoven.

2.Maximizeevendistributionofheatforthepan.

Optimizeacombinationofconditions1and2whereweightspand(1-p)areassignedtoillustratehowtheresultsvarywithdifferentvaluesofW/Landp.

MaximizingthenumberofpansinarectangularovenisequivalenttotheproblemthathowtoarraymoredifferentgeometricfigureswiththesameareaofAinarectangle.ThroughtheLeiHuangetc[4],weuseRectangularPackingAlgorithmtosolveit.

Tomaximizeevendistribution,figuringouttheheatdistributioninthepansfordifferentshapesiscrucial.ApreliminaryresearchofheatconductionhasbeencarriedoutbyFranklinC.daSilvaetc[1]:

TheTwo-DimensionalHeatConductionEquationisusuallyusedtodescribe

theheatconductionprogress.

PDEToolboxinMATLABcanbeusedtosimulatetheheatpartialdifferential

equation,especiallyfortheconductionequation.

Sowecanapplytheresearchabovetodevelopthemathematicmodelofheatdistributionandsimulateit.

VariablesandAssumption

Variables

Variable

u(x,y,t)

Q

Tw

Tf

S

ATable1.VariablesusedinthemodelDefinitionthetemperatureoftheanypointx,yofpanatthetimet( C)theheattransferringtothepanintheunitareaandunittime(W/m2)theouteredgetemperatureofthepan( C)theairtemperatureintheoven( C)theareaoftherectangularoven(cm2)theareaofthepan(cm2)

anindextodescribethedegreeoftheuniformityofheatdistribution

anindextomeasuretheoptimizenumberandtheevendistributioncomprehensively

美国大学生数学建模竞赛

GeneralAssumptions

Theheatcanonlytransfertothepanfromitsouteredgethroughtheair.Sincethefoodplacedonitpreventstheheatfromconductingtoit,thisisareasonable.Thetemperatureinovenisevensincetheairisflowing.Thatthereisonlyonekindofpansintheovens.Initiallytherearetworacksintheoven,evenlyspaced.Wesupposethatthe

temperatureandheatareequivalentandconstant,sowejustconsiderateonerackandtheotheroneisthesamewithit

Theratiooftheovenplane’swidthandlengthisW/L.

EverypansharesthesameareaofA.

Thedatawecitedinthemodelsaretrue.TheareaoftheovenisS 750cm2,andtheratiobetweenwidthandlengthisW/L 22:34.Moreover,theareaofpanisA 100cm2.[2]

TheMathematicalModelofHeatDistribution

Definitions

TheouteredgeandouteredgeandinnerofthepansareillustratedintheFigure1.

Figure1.Theillustrationofdifferentedges.

1.Overview

Firstly,weassumethatthetemperatureinsidetheovenisaconstantandthe

motionstateoftheinternalairisstable.Asaresult,weignoretheinfluencefromtheinternalenvironmentoftheovenwhenheatconductinginthepan.Theairflows

circularlyinsidetheoventoensurethesteadytemperature(intheoven).Accordingtothis,weassumedthatthereisnocertainflowdirectionofair,andtheflowvelocitytowardsanydirectionisconsistent.

Decomposetheairflowintothreedirectionsparallelingtothex,y,andz-axis.Whentakingthepanasathinplane,thereisnoheattransferringtothepanfromthez-axis’directionbecauseofthefoodplacedonit.Sotheprocessofheattransfercanbedescribedbyatwo-dimensionalheatconductionequation.

Theschematicdiagramoftwo-dimensionalheatconductionprocessshowsin

Figure2.

美国大学生数学建模竞赛

HeatdirectionConductionHeatConduction

direction

HeatConduction

direction

Figure2.Diagramoftwo-dimensionalheatconductionprocess

2.Two-dimensionalHeat-conductionEquation

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