Manufacturing2.810 JohnLewandowski

HeatTransferQuizReviewSheet

OverviewForthosewhohaven’ttakenheattransfer(meincluded),theseequationscanlookprettyscaryifyoudon’twalkthroughstepbystep.HereIwillshowyouthatthemajorityofthedifficultyanddifferencesbetweenallofthetypesandcasesjustboilsdowntothreeaspects:

1. Equilibriuma. Alwaysstartwith𝑞"# = 𝑞%&'andtrytosolveforatimeorotherquantity

2. Boundaryconditionsa. Mostofthetimeweareinterestedinoneparticularboundaryconditionandthe

twotypesofheattransferintoandoutoftheboundary3. Dimensionlessnumbers

a. Thinkaboutwhythesedimensionlessnumbersexistandrelatethembacktothemanufacturingprocess

Ingeneral,itisalsoimportanttounderstandhowmaterialschangetheirphase.Thereisagoodhomeworkquestionthatdiscusseshowtocalculatetheenergyrequiredtoheatamaterialaswellasthelatentheatoffusiontochangeitsphase.Youcanplotthoserelationshipscalculationsdirectlyontothesediagrams.

KeyParameters

𝑞 = ℎ𝑒𝑎𝑡 = [𝐽]𝑞𝐴 = ℎ𝑒𝑎𝑡𝑝𝑒𝑟𝑢𝑛𝑖𝑡𝑎𝑟𝑒𝑎 = [

𝐽𝑚7]

𝑃 = 𝑝𝑜𝑤𝑒𝑟 = 𝑊

Manufacturing2.810 JohnLewandowski

𝑐= = ℎ𝑒𝑎𝑡𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 =𝐽

𝑘𝑔𝐾

𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = [𝑘𝑔𝑚E]

Iwilladdmoretothesesoon

KeyFormulasRadiative

𝑞"# = 𝑞%&'

𝑞"# = 𝑃FGH=𝑎Forradiativeheating,wejusthavetheheatcomingfromthepowerofthelampandtheabsorptioncoefficientofthesheet.Aspowerorcoefficientincreases,sodoestheheatin.

𝑞%&' = 𝑚𝐴 𝑐=

𝑑𝑇𝑑𝑡

∂Thisisjustsayingthattheheatcomingintothesystemisbeingusedtoheatupthematerialassometemperatureriseratebasedonitsheatcapacitanceanddensity.Wecansettheseequalnow.

𝑞"# = 𝑞%&' = 𝑚𝐴 𝑐=

𝑑𝑇𝑑𝑡 = 𝑃FGH=𝑎

𝑑𝑡 =𝜌ℎ𝑐=𝑃FGH=𝑎

𝑑𝑇

Thenjustthinkabouttheprocess.Wearegoingfromtzerototimeofheatingandbetweentheinitialandfinaltemperature.Easy!

𝑑𝑡'KLMN

O=

𝜌ℎ𝑐=𝑃FGH=𝑎

𝑑𝑇PQ

PR

𝑡STG'U𝜌ℎ𝑐=𝑃FGH=𝑎

(𝑇W − 𝑇O)

Alsothinkabouttherelationships.Asthethickness(h)increases,timetoheatincreases.Asthepowerorabsorptioncoefficientgoesup,thetimetoheatdecreases.Alwaysdothisanalysisasacheck!

Manufacturing2.810 JohnLewandowski

Conduction

𝑞"# = 𝑞%&'Fourier’slawtellsushowheatisbeingconductedthroughthematerialandwecanassumeonedimensionalheattransferintheplatesowegetridofthethree-dimensionaltemperaturegradient.

𝑞"# = −𝑘∆𝑇

𝑞"# = −𝑘𝑑𝑇𝑑𝑥

Thenwecanmodelthetemperatureacrosstheboundaryofthesystem

𝜕𝑇𝜕𝑡 =

𝑘𝜌𝑐=

𝜕7𝑇𝜕𝑦7 = 𝛼

𝜕7𝑇𝜕𝑦7

Wedon’tneedtogofartherthanthis,theyhelpussolveitanalyticallyandthengiveustheapproximation.

𝑡^%%F =ℎ7

𝜋7𝛼 𝑙𝑛4𝜋

𝑇HTF' − 𝑇bGFF/H%Fd𝑇TeT^'"%# − 𝑇bGFF/H%Fd

𝑡^%%F =(ℎ/2)7

𝛼

Manufacturing2.810 JohnLewandowski

It’ssoimportanttounderstandhowthisgraphrelatestotheBiotnumberandtheFouriernumberandthedimensionlesstemperaturevalue.Thegraphaboveisshowingthetemperaturedistributionatmultipleinstancesoftime.Notethatatthetopthereis𝑡O,rightwhenthemoltenmaterialentersthemoldandthetemperaturedistributionisconstantacrosstheentirepart.Thendependingonitsmaterialproperties,it’sgoingtodeterminehowquicklyitcools(thecurvatureofthelines)becausethewalltemperatureismuchlessthanthetemperatureofthematerial.Dependingonthetemperaturesselected,it’sgoingtodeterminetheheightofthisdiagram.Thethicknessofthepartisthewidthofthediagram.Thecenterlineofthepartisinthemiddleandasyoumoveacrossthepart,itrelatestothedifferent6or9boxesthatmakeuptheBiotnumbergraphsbelow.TheBiotnumberisyourmaterialpropertydimensionlessnumber,whichistellingyouwhatthatcurvaturelookslikeandiswhyeachlineinsidetheboxhasadifferentslope.Thenthex-axisisyourFouriernumberandthatisanindicationofhowlongyouhavewaitedandwhichtimelineyouareonabove.They-axisisthetemperaturewhicheitheramplifiesorde-amplifiestheheattransferprocessbyelongating/shrinkingthatgraphabove.

Manufacturing2.810 JohnLewandowski

ConvectionConvectiveheattransferisveryeasytoremember,itjusttakesintoaccounttheareaoftheobject,theheattransfercoefficient,andthetemperaturedifference.

𝑞 = ℎ𝐴(𝑇 − 𝑇g)

𝑞𝐴 = ℎ(𝑇 − 𝑇g)

Thewaythatyouwouldusethisismostlikelyforapartnexttoambienttemperature(eithercastingormaybethermoforming)tosettheequilibriumagain.

𝑞"# = 𝑞%&'Theheatenteringthemoldisonlycomesfromtheheatoffusionofsolidifyingmetal.

𝑞𝐴 = −𝜌𝐻

𝑑𝑆𝑑𝑡

SettheseequalandsolveafterintegrativethesimpledSanddt.

Manufacturing2.810 JohnLewandowski

𝑆 = ℎ(𝑇 − 𝑇g)𝑡

𝜌𝐻

𝑡 = 𝜌𝐻

ℎ(𝑇 − 𝑇g)𝑆

𝑡 = 𝜌𝐻

ℎ(𝑇 − 𝑇g)𝑉𝐴

𝑡 = 𝐶𝑉𝐴

S(lengthdimension)basicallyturnsintoavolume/areatermandtheentirecoefficientreducesintotheCthatyouseeinthecastinglecture.Flemingswalksthroughallofthedifferentcasesinthebelowexamplesdependingonthematerialthatisnexttothewall.

Manufacturing2.810 JohnLewandowski

Manufacturing2.810 JohnLewandowski

KeyReadingsFlemingsSolidificationNotes(DropboxLinkinsideCasting)isamust!Itshowsallofthedifferentcases.BeabletoexplainFigures1-3,1-6,and1-9alongwiththecorrespondingequations.