2013年美国大学生数学建模大赛B题获奖论文

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22599

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A

2013

MathematicalContestinModeling(MCM/ICM)SummarySheet

(Attachacopyofthispagetoyoursolutionpaper.)

HeatRadiationinTheOven

Heatdistributionofpansintheovenisquitedifferentfromeachother,whichdependsontheirshapes.Thus,ourmodelaimsattwogoals.Oneistoanalyzetheheatdistri-butionindifferentovensbasedonthelocationsofelectricalheatingcubes.Further-more,aseriesofheatdistributionwhichvariesfromcircularpanstorectangularpanscouldbegoteasily.Theotheristooptimizethepansplacing,inordertochooseabestwaytomaximizetheevenheatandthenumberofpansatthesametime.

Mathematicallyspeaking,oursolutionconsistsoftwomodels,analyzingandoptimi-zing.Inpartone,ourwhole-localapproachshowstheheatdistributionofeverypan.Firstly,weusetheStefan-BoltzmannlawandFouriertheoremtodescribetheheatdistributionintheairaroundtheelectricalheatingtube.Andthen,basedonplanein-terceptmethodandsimplifiedMonteCarlomethod,theheatdistributionofdifferentshapesofpansisobtained.Finally,weexplainthephenomenonthatthecornersofapanalwaysgetoverheatedwithwaterwavesstirringbyanalogy.Inparttwo,ourdiscretize-convertapproachoptimizestheshapeandnumberofthepans.Aboveall,wediscre-tizethesidelengthoftheoven,sothatthenumberandtheaverageheatofthepansvarylinearly.Intheend,theabstractweightPisconvertedintoaspecificlength,inordertoreachacompromisebetweenthetwofactors.

Specially,wecreateauniquemethodtoconvertthevariablesfromthewholespacetothelocalsection.Thespecialmethodallowsustodrawtheheatdistributionofeverysinglesectionintheoven.Thealgorithmwecreatedoesagreatjobinflexibility,whichcanbeappliedtoallshapesofpans.

Typeasummaryofyourresultsonthispage.Donotincludethenameofyourschool,advisor,orteammembersonthispage.

HeatRadiationinTheOven

Summary

Heatdistributionofpansintheovenisquitedifferentfromeachother,whichdependsontheirshapes.Thus,ourmodelaimsattwogoals.Oneistoanalyzetheheatdistri-butionindifferentovensbasedonthelocationsofelectricalheatingcubes.Further-more,aseriesofheatdistributionwhichvariesfromcircularpanstorectangularpanscouldbegoteasily.Theotheristooptimizethepansplacing,inordertochooseabestwaytomaximizetheevenheatandthenumberofpansatthesametime.

Mathematicallyspeaking,oursolutionconsistsoftwomodels,analyzingandoptimi-zing.Inpartone,ourwhole-localapproachshowstheheatdistributionofeverypan.Firstly,weusetheStefan-BoltzmannlawandFouriertheoremtodescribetheheatdistributionintheairaroundtheelectricalheatingtube.Andthen,basedonplanein-terceptmethodandsimplifiedMonteCarlomethod,theheatdistributionofdifferentshapesofpansisobtained.Finally,weexplainthephenomenonthatthecornersofapanalwaysgetoverheatedwithwaterwavesstirringbyanalogy.Inparttwo,ourdiscretize-convertapproachoptimizestheshapeandnumberofthepans.Aboveall,wediscre-tizethesidelengthoftheoven,sothatthenumberandtheaverageheatofthepansvarylinearly.Intheend,theabstractweightPisconvertedintoaspecificlength,inordertoreachacompromisebetweenthetwofactors.

Specially,wecreateauniquemethodtoconvertthevariablesfromthewholespacetothelocalsection.Thespecialmethodallowsustodrawtheheatdistributionofeverysinglesectionintheoven.Thealgorithmwecreatedoesagreatjobinflexibility,whichcanbeappliedtoallshapesofpans.Keywords:MonteCarlo

discretization

thermalradiation

section

heatdistribution

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Introduction

Manystudiesonheatconductionwastedplentyoftimeinsolvingthepartialdifferentialequations,sinceit’sdifficulttosolveevenforcomputers.Weturntoanotherwaytoworkitout.Firstly,westudytheheatradiationinsteadofheatconductiontokeepawayfromthesophisticatedpartialdifferentialequations.Then,wecreateauniquemethodtoconverteveryvariablefromthewholespacetosection.Inotherwords,weworkeverythingoutinheatradiationandconvertthemintoheatcontradiction.Assumptions

Wemakethefollowingassumptionsaboutthedistributionofheatinthispaper.·Initiallytworacksintheoven,evenlyspaced.

·Whenheatingtheelectricalheatingtubes,thetemperatureofwhichchangesfromroomtemperaturetothedesiredtemperature.Ittakessuchashorttimethatwecanignoreit.

·Differentpansaremadeinsamematerial,sotheyhavethesamerateofheatconduction.

·Theinnerwallsoftheovenareblackbodies.Thepanisagraybody.Theinnerwallsoftheovenabsorbheatonlyandreflectnoheat.

·Theheatcanonlybereflectedoncewhenreboundedfromthepan.HeatDistributionModelOurapproachinvolvesfoursteps:

·UsetheFouriertheoremtocalculatethelossenergywhenenergybeamsarespreadinthemedium.Sowecangettheheatdistributionaroundeachelectricalheatingtube.Theheatdistributionoftheentirespacecouldbegowheretheheatoftwoelectricalheatingtubescrosstogether.

·Whendifferentshapesofthepansareinsertedintotheoven,theheatmapoftheentirespaceiscrossedbythesectionofthepan.Thus,theheatmapofeverysinglepanisobtained.

Establishasuitablemodeltogetthereflectivityofeverysinglepointonthepanwith·

thesimplifiedMonteCarlomethod.Andthen,afinalheatdistributionmapofthepanwithoutreflectionlossisobtained.

·Arealisticconclusionisdrawnduetotheresultsofourmodelcomparedwithwaterwavepropagationphenomena.

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Firstofall,thepaperwillgiveadescriptionoftheinitialenergyoftheelectricalhea-tingtube.Weseeitasablackbodywhoreflectsnoheatatall.Electromagneticknow-ledgeshowsthatwavelengthoftheheatraysrangesfrom10−1umto102umasshownbelow[1]:

Figure1.

Figure2.

WeapplytheStefan-Boltzmann’slaw[2]whosesolutionis

c1λ−5

Ebλ=c2/(λT)e−1∞∞

c1λ−5

Eb=∫Ebλdλ=∫c2/(λT)dλe−100

WhereEbmeanstheabilityofblackbodytoradiate.c1󰀀andc2areconstants.Obviously,,theinitialenergyofablackbodyisEb0=3.2398e+012(w×m2).CombineFigure1withFigure2,weintegrate(1)fromλ1toλ2togettheequation

asfollow:

λ2

(1)(2)

Eb(λ1−λ2)=∫Ebλdλλ1

(3)

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Figure3.

FromFigure3,itcanbeseenhowthepowerofradiationvarieswithwavelength.Secondly,basedontheFouriertheorem,therelationbetweenheatandthedistancefromtheelectricalheatingtubesis:

dt(4)Q=−λSdxWhereQisthepowerofheat(J/s=W),Sistheareawheretheenergybeam

dtradiates(m2),representsthetemperaturegradientalongthedirectionofenergy

dxbeam.[3]

Itisknownthattheenergybecomesweakerasthedistancebecomeslarger.Accordingtothefactweknow:

dQ(5)ρ=

dxWhereρistherateofenergychanging.

Weassumethatthedesiredtemperatureofelectricalheatingtubeis500k.Withthetwoequations,thedistributionofheatisshownasfollow:

Figure4.(a)Figure4.(b)

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Inordertodrawthemapofheatdistributionintheoven,weuseMATLABtoworkonthecomplicatedalgorithm.TherelationbetweenthepowerofheatandthedistanceisshowninFigure4(a).TherelationbetweentemperatureanddistanceispresentedinFigure4(b).ThespreadingdirectionofenergybeamispresentedinFigure5.

Figure5.

Theshapeofelectricalheatingtubeisirregular.Theheatdistributionofasingleelectricalheatingtubecanbedrawin3DspacewithMATLAB.ThepictureisshowninFigure6.Aftersuperimposing,thetotalheatdistributionoftwotubesisshownbelowinFigure7andFigure8.

Figure6.Figure7.

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Figure8.

Thepicturesaboveshowtheenergyinanovenwithnopan.WeputinarectangularpanwhoseareaisA,andinterceptthemapswithMATLAB.TheresultisshowinFigure9.

Figure9.

Figure10.

Putinacircularpantointerceptthemaps,whoseareaisA,also.ThedistributionofheatisshowninFigure11.

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Figure11.

Whenputinapanintransitionshape,whichisneitherrectangularnorcircular.TheareaofitisA,also.Theheatdistributiononsuchapanisshownasfollow:

Figure12(a)

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Figure12(b).

Figure13.

Next,learningfromtheMonteCarlosimulation[4],amodelisestablishedtoget

obtainthereflectivity.Wegeneratearandomnumberbetween0and1todetermineiftheenergybeamoncertainpointisreflected.

•Firstly,todemonstratethequestionbetter,weconstructasimplemodel:

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Figure14.

Whereθistheviewinganglefromelectricalheatingtubetothepan.R=theproportionofthebeamsradiatedtothepan.

•Whatismore,weassumethetotalbeamisM1.Ideally,thenumberofabsorptionis

θ.Then,eachelementofthepanisseenasagridpoint.EachgridpointcanM1×360

θgeneratea-M1×-random-numbervectorbetween0and1inMATLAB.

360•AfterMATLABsimulating,thenumberofbeamsdecreasedbyM2,duetothe

360×M2

reflection.Sowedefineaprobabilityρ=todescribethenumberofbeams

M1θreflected.Theconclusionis:

•Ifρ≤R,theenergybeamisabsorbed.•Ifρ>R,theenergybeamisreflected.[5]

Basedontheanalysisabove,ourmodelgetafinalresultofheat-distributiononthepanasshownbelow:

θis360

Figure15(a)

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Figure15(b).

Theconclusionisknownthattheclosertheshapeofpansistocircle,themoreevenlytheheatisdistributed.Moreover,thephenomenonthatthecornersalwaysgetoverheatedcanbeexplainedbywaterwavepropagationindifferentcontainers.

Whenthereisafluctuationinthecenterofthewater,therippleswillfluctuateandspreadinconcentriccircles,asshowninFigure16.Thefluctuationstirswavesupwhencontactingtheborderlines.Comparedwiththewaveswithoneboundary,thewavesincornermakeahigheramplitude.

Thethermalconductiononthepanisexactlytheinverseprocessofthewavespropagation.Therangeofthermalmotionismuchsmallerthanitontheside.That’swhythecornersiseasytogetoverheated.

Inordertomaketheheatevenlydistributedonthepan,thesidesofthepanshouldbeasfewaspossible.Therefore,ifnothingisconsideredabouttheutilizationofspace,acirclepanisthebestchoice.

Figure16,thewaterwavespropagation[6]

AccordingtotheanalysisaboveandFigure7,thephenomenonshowsthattheheatconductionissimilartowaterwavespropagation.Soitisprovedthatheat

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concentratesinthefourcornersoftherectangularpan.TheSuperPanModelAssumptions

•Thewidthoftheoven(W)is100mm,thelengthisL.•Therearethreepansatmostinverticaldirection.•Eachpan’sareaisA.

Thefirstpart.

Calculatethemaximumnumberofpansintheoven.Differentshapesofpanhavedifferentheatdistributionwhichaffectsthenumberofpans,judgingfromtheprevioussolution.Accordingtotheconclusioninthefirstmodel,theheatisdistributedthemostevenlyonacircularpanratherthanarectangularone.However,therectangularpansmakefulleruseofthespacethespacethancircularones.Bothfactorsconsidered,apolygonalpanischosen.

Acirclecanberegardasapolygonwhosenumberofboundariestendstoinfinity.Exceptforrectangle,onlyregularhexagonandequilateraltrianglecanbecloselyplaced.Becauseoftheedgesofequilateraltriangle,heatdissipationisworsethanrectangle.So,hexagonalpansareadoptedafterallthediscussion.

Consideringthegapsnearboundaries,weplacethehexagonalpanscloselyattachedeachotheronthelongsideL.Therearetwokindsofprogramsasshownbelow.

Program1.

Program2.

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Obviously,Program2isbetterthanProgram1whenconsideringspaceutilization.Soscheme1isadopted.

Then,designasizeofeachhexagonalpantomakethehighestspaceutilization.Withtheaimofutilization,hexagonalpanshastobeplacedcontactcloselywitheachotheronbothsides.Itisnecessarytoassumeaaspectratiooftheoventoworkoutthenumberofpans(N).

Assumethatthesidelengthofaregularhexagonisa,thelength-widthratiooftheovenisλand∆Listheincrementindiscretization.Becausethenumberofpanscannotchangecontinuouslywhenn∈(m,m+1),m=1,2,3⋅⋅⋅,theequationswouldbeasfollows.

⎧W⎪5=a⎪

⎪0⎪L+k⋅∆L⎡L+k⋅∆L⎤⎪0≤−⎢<1(6)⎨⎥W⎣W⎦⎪

⎪W=λ⎪

⎪L+k⋅∆L⎪N0=8,L0=W⎪

⎡⎤⎪

⎢L−W⎥⎪

⎥⎪n=⎢3⎢⎥⎪⋅a⎢⎥⎪⎣2⎦⎩

n−1⎧

N=N+3⋅+1n=2k−1,k=1,2,3⋅⋅⋅0⎪⎪12Result:⎨

⎪N=N+3⋅nn=2k,k=1,2,3⋅⋅⋅20⎪⎩2

WhereN1representsthenumberofpanswhennisodd,N2representsthenumberofpanswhenniseven.Thespecificnumberofpansisdependedonthewidth-lengthratioofoven.

Thesecondpart.

Maximumtheheatdistributionofthepans.Wedefinetheaverageheat(H)astheratiooftotalheatandtotalareaofthepans.Aimingtogetthemostaverageheat,wesetthewidth-lengthratiooftheovenλ.Spaceutilizationisnotconsideredhere.AconclusioniseasytodrawfromFigure8thatasquareareaintheovenfrom150mmto350mminlengthsharesthemostheatevenly.Sothepansshouldbe

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placedmainlyinthisarea.Frommodel1weknowthatthecornersoftheovenareapttogatherheat.Besides,fourmorepansareaddedinthecornerstoabsorbmoreheat.Becauseheatabsorbingistheonlyaim,thereisnoneedtoconsiderspaceutilization.Circularpanscandistributeheatmoreevenlythananyothershapeduetomodel1.SocircularpansareusedinFigure17.

Figure8.

Figure17.

Wesettheheatofthepansinthemostheatedarea(themiddlerow)asQ.Pansinthecornersreceivemoreheatbutuneventheoretically.AndthesquareofthefourpansinthecornersissosmallcomparedwiththetotalsquarethatwesettheheatofthefourasQtoo.Whenthelengthofoven(L)increases,thenumberofpansincreasestoo.Itmakesthesquareofthegapsbetweenpansbigger,meanwhile.Ifeachpanhasasameradius(r)andsquare(A),theequationaboutaverageheat,length-widthratioandnumberofpanswouldbe(7).

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⎧13r=⋅⋅W⎪

45⎪

⎪A=π⋅r2⎪

⎪0⎪∆L=2⋅r⎪

L+k⋅∆L⎡L+k⋅∆L⎤⎪0≤−⎢⎨⎥<1WW⎣⎦⎪

⎪W=λ⎪

L+k⋅∆L⎪

⎪N0=7;L0=W⎪

⎪n=⎡L−W⎤

⎢2⋅r⎥⎪⎣⎦⎪

⎪N=N0+n⎪⎩k=1,2,3...

(7)

Herewegetthemostaverageheat(H):

400QH=⋅2

9πWThethirdpart.

Wediscussedtwodifferentplansinthepreviouspartsofthepaper.Oneisaimedtogetthemostaverageheat,whiletheotheraimedtoplacethemostpans.Thetwoplansarecontradictorywitheachother,andcannotbeachievedtogether.

Firstly,theweightofplan1isPandtheweightofplan2is1−P.Obviously,thiskindoptimizationhasdifficultyinsolvingandunderstanding.Soweturntoanotherwaytomakeitaeasierandlinearquestion.IthasbeensetthatthewidthoftheovenisaconstantWandthereshouldbethreepansatmostinverticaldirection.WemaketheweightPaproportionofthetwoplans.ThusthetwoplanscouldbeachievedtogetherduetoproportionPand1−P,asshowninFigure18.

Figure18.

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Asbeentoldinmodel1,thecornershaveahighertemperaturethanotherpartsoftheoven.Soplan1isusedindistrict1(inFigure10)andplan2isusedindistrict2(inFigure10).Abettercompromisecouldbereachedinthisway,asshowninFigure19.

Figure19.

EverypanhasasquareofA.Radiusofcircularonesisr.Sidelengthofregularhexagonisa.

π⋅r2=

332

⋅a2

⇒a:r=1.1(8)

Basedontheequation(8),ifthepansareplacedasshowninFigure19,regularhexagonsareplacedfullofdistrict1,thecircularoneswillbeplacedbeyondtheborderline.Ifthecircularonesareplacedfullofdistrict2,therewillbemoregapsindistrict1,whichwillbewasted.SowechangeourplanofplacingpansasFigure20.

Figure20.

Thenumberofcircularonesdecreasesbytwo,butthespaceindistrict1isfullyused,andnopanwillbeplacedbeyondtheborderline.

WeassumethatPisbiggerthan1−P,sothat,theheatindistrict1willbefullyused.Bysimplecalculating,weknowthattheratiooftheheatabsorbedincircularpan(H1)andinregularhexagon(H2)is1.2:1.

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Figure21.

So,basedonthepansplacingplan,aequationonheatcanbegotasfollow:

⎧3a≈r=x;⎪2⎪⎪33

⋅a=π⋅r2⎪A=2⎪⎪W0<<1⎪

L⎪

⎪L=L0+k⋅∆L⎪∆L=x⎪⎪L+k⋅∆L⎡L+k⋅∆L⎤

−⎢⎨0≤⎥<1WW⎣⎦⎪

⎪W=λ⎪

L+k⋅∆L⎪

⎪N0=9,L0=W⎪

⎪n=⎡P⋅L−W⎤⎪1⎢x⎥⎣⎦⎪

⎪n=⎡(1−P)⋅L−W⎤⎪2⎢x⎥⎣⎦⎪

⎩k=1,2,3...

(9)

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Resolution:

n1−1⎧

N=N+2⋅n+3⋅+1(n1=2k−1)02⎪1

2

⎪

⎪N=N+2⋅n+3⋅n1

(n1=2k)12

⎪22⎪n−1⎪(5+1⋅3)⋅Q+1.2Q⋅(4+2⋅n2)⎪2⎨H1=

N1⋅A⎪

⎪n1

(5+⋅3)⋅Q+1.2Q⋅(4+2⋅n2)⎪

2⎪H2=

N2⋅A⎪

⎪k=1,2,3...⎪⎩

(10)

N1andH1meansthenumberofpansandaverageheatabsorbedwhennisodd.N2andH2meansthenumberofpansandaverageheatabsorbedwhenniseven.Forexample:

(1)Whenλ=0.37,P=0.6:

QN=16,H=1.075.Thebestplacingplanis:

A(2)Whenλ=0.37,P=0.7:

QN=18,H=1.044⋅.Thebestplacingplanis:

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(3)Whenλ=0.58,P=0.6：

QN=12,H=1.067⋅.Thebestplacingplanis:

AAconclusioniseasytodrawthatwhentheratioofwidthandlengthoftheoven(λ)isaconstant,thenumberofpansincreaseswithanincreasingP,buttheaverageheatdecreases(example(1)and(2)).WhentheweightPisaconstant,thenumberofpansdecreaseswithanincreasingλ,andtheaverageheatdecreasesalso.So,theactualplanshouldbebaseonyourspecificneeds.Conclusion

Inconclusion,ourteamisverycertainthatthemethodwecameupwithiseffectiveinheatdistributionanalysis.Basedonourmodel,themoreedgesthepanhas,themoreevenlytheheatdistributeon.Withthediscretize-convertapproach,weknowthatwhentheratioofwidthandlengthoftheoven(γ)isaconstant,thenumberofpansincreaseswithanincreasingP,buttheaverageheatdecreases.WhentheweightPisaconstant,thenumberofpansdecreaseswithanincreasingγ,andtheaverageheatdecreasesalso.So,theactualplanshouldbebaseonyourspecificneeds.

Strengths&Weaknesses

Strengths

.•DifficultiesAvoidedAvoided.

Inmodel1,weturntoanotherwaytoworksimulatetheheatdistributioninsteadofworkonheatconductiondirectly.Firstly,wesimulateheatradiationnotheatconductiontokeepawayfromthesophisticatedpartialdifferentialequations.Then,wecreateauniquemethodtoconverteveryvariablefromthewholespacetosection.Inotherwords,weworkeverythingoutinheatradiationandconvertthemintoheatcontradiction.

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•ClosetoReality.

Ourmodelconsidersboththethermalradiationandsurfacereflection,whichisrelativelyclosetotheactualsituation.

•FlexibilityProvided.

Ouralgorithmdoesagreatjobinflexibility.Theheatdistributionmaponsectionsareinterceptedfromtheheatdistributionmapsoftheentirespace.Allshapesofsectionscanbeusedinthealgorithm.Theheatdistributioninthewholespaceisgeneratedbasedonthelocationoftheelectricalheatingtubesandthedecaycurveoftheheat,whichcanbemodifiedatanytime.

•Innovation.

Basedonourmodel,thespaceofanovencanbedividedintosixpartswithdifferentheardistribution.Inordertomakefulluseoftheinnerspace,weinventanewpanwhichallowsuserstocooksixdifferentkindsoffoodatsametime.Anadvertisementispublishedintheendofthepaper.

Weaknesses

’sThermalConductivityIgnored.•PanPan’

Theheatcomesfromnotonlytheelectricalheatingtubes,butalsoheatconductionofthepansthemselves.Butthepan’sthermalconductionisignoredinthemodel,whichmaycauselittleinaccuracy.

.•ThermalConductivityofElectricalHeatingTubesIgnoredIgnored.

itisassumedthattherearetwoelectricalheatingtubesintheovenandplacedinaspecificlocation.Theinitialtemperatureofthetubesisadesiredconstanttemperature.Inotherwords,thetimeelectricalheatingtubesspendtoheatingthemselvesisignored.Thesimplificationcancausesomeinaccuracy.

.•Linearsimplificationsimplification.

Inmodel2,thelengthoftheovenisdiscretized,sothatthenumberofpanswillchangeslinearly.calculatingthroughsimpleintegerlinearmethod.Thiswillleadtotheresultofourmodelisnotaccurateenough.Application

Wehavediscussedtheheatdistributionintheoveninmodel1.Theheatdistributionisshowninfigure1andfigure2.

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Figure1

Figure2

Asshown,theedgesoftheovenaredistributedthemostheat.Areasonbothsidesofthe,isdistributedtheleastheat.Whilethemiddleareaabsorbslittlelessthantheedges.So,wecanseparatetheovenareaintosixparts,asshownbellow.

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Part1andpart2aredistributedtheleastheatandlocatedthefurthestfromtheheatsource(theelectricalheatingtubeslocateonthebottomoftheoven).Sothesetwopartsabsorbtheleastheat.Part3andpart4aredistributedtheleastheatbutlocatingthenearesttotheheatsource.Part5locatedfarfromthebottombutdistributedthemostheat.Sosimply,weregardtheheatofpart3,part4andpart5asthesame.Part6isdistributedthemostheat,andlocatingnearesttothebottom.So,theheatpart6absorbsisthemostintheoven.

Basedonourconclusionabove,weinventtheiPan,anewcombinedpan,whichcanbakethreekindsoffoodatthesametime.Forexample,onewantstohavealittlebread,piecesofsausage,achickenwingandapizzaforlunch.Hewillhavetowait30minutesatleastforhislunch,ifhejusthasoneoven.AstheChinesesayinggoes,‘Bearpawsandfishnevercometogether’.

ByusingiPancansolvetheissueforhim,hecouldputthebreadinpan1,pizzainpan2,sausageinpan5andchickenwinginpan6,andpoweron.Thus,hecanhavehisdeliciouslunchinatleast10minutes.So,bearpawsandfishcometogether.WemakeanadvertisementforBrownieGourmetMagazineintheendofthepaper.AdvertisingSheets

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References

[1]HeatRadiation,http://wenku.baidu.com/view/f5ed1619cc7931b765ce1599.html,Page.4

[2]G.S.Ranganath,Black-bodyRadiation,

http://link.springer.com/article/10.1007%2Fs12045-008-0028-7?LI=true#,February,2013

[3]KaiqingLu,TheChemicalBasisofHeatTransfer,JournalofHigherCorrespondenceEducation(NaturalSciencesEdition),Vol.3:p.33,1996

[4]MarkM.Meerschaert,MathematicalModeling(ThirdEdition),China:ChinaMachinePress,May.2009

[5]JianzhongZhang,MonteCarloMethod,MathematicsinPracticeandTheory,

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Vol.1p.28,1974

[6]Shallowwaterequations,http://en.wikipedia.org/wiki/Shallow_water_equations