CHE342Fall08GroupProject
ThermalConductivityExperiments(DeterminationoftheLengthofanAluminumRod)
FormalReportOctober‐December,2008EricChuang
VincentHusaini
DeandreReagins
MarcSinger
Group#8December10,2008
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FOREWORDThisChE342Fall2008groupprojectentailsdesigninganoriginalexperimentthatissuitableforahighschoolteachertousefordemonstratingaheattransferconcepttohis/herclass.Thegoaloftheprojectistousethisexperimenttoattracthighschoolstudentstochemicalengineering.Theexperimentaldesignmustbefeasibleanditscostformaterialsandsetupcannotexceed$25.Theexperimentmustbeeasilysetupinaclassroomenvironmentandtakenomorethan15minutestoexecute.Thepurposeofthisreportistopresenttherationaleanddesignofourexperiment,ourresults,andconclusionsSUMMARYSeveralideaswerediscussedovertwomeetingswithourgroup,includinganapparatusthatproduceshydrogengaswhichisblownupattheendoftheexperiment.Thehydrogengasapparatuswasnotapproved;howeveranapparatusthatcanbeusedtoaccuratelydeterminethelengthofametalrodusingheattransferprincipalswasapprovedandwassuccessfullydemonstratedinfrontofagroupofhighschoolstudents.Adiagramoftheapparatusisshownbelow.
Figure1:Apparatusdiagram
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RATIONALEWe desired to quantitatively and qualitatively illustrate a basic concept of heat transfer tostudentsusinganexperimentalsetupthatwaseasilyrepeatableandeasytoduplicateinahighschool classroom setting. Additionally we wanted to exhibit to the students a practicalapplication of the concepts introduced in our experiment. As a group we decided that anexperimentcomparingdifferentthermalconductivityvalueswouldaccomplishourgoals.Froma quantitative standpoint we planned to compare the numerical k‐values for each type ofmaterialgraphicallyillustratethetemperaturechangeforagivensizeofmetaloveraspecifiedamount of time. Quantitatively we planned to use thermometers to track and record thetemperature change for each of our demonstrations. Practical application was also animportantaspectofour rationale for thisexperiment.Knowingthat it ispossible tocalculatethe length of a piece of pipe given the material properties and the overall change intemperature we designed our experiment in such as way that students would be able tounderstandthiscalculationconceptually.EXPERIMENTALSETUP/MEASUREMENTS
Figure2:ExperimentalSetup
1. Pour~175mLofwaterintoa250mLbeaker.Placethebeakerontoahotplateandheat
thewatertodesiredtemperature.Recordthewatertemperature.2. DrillorpierceasmallholeintooneoftheStyrofoamcylinders.Theholeshouldbesized
sothatthemetalrodfitssnuglywithintheStyrofoaminsulation.3. InsertthemetalrodintothecenteroftheStyrofoamcylinder.Approximately3mmof
metalshouldbeexposedbeneaththeStyrofoamcylinder.Usethedigitalthermometerto record the temperature at the top of themetal rod. Repeat this process for eachmetalrodthatyouhave.
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4. Withtherodinserted,carefullyplacetheStyrofoamcylinderintotheboilingwaterbathsothat6mmofthecylinderissubmergedbeneaththewaterline.Clampthecylinderinplaceusingoneoftheclampsandthestand
5. Placethedigitalthermometerontothetopofthemetalrod.Taketemperaturereadingsatregulartimeintervals.
Theideaistohaveahotwaterbathwiththetemperaturecarefullymonitoredandimmerseaninsulatedmetalrod(therodusedinthisexperimentisaluminum,butcanbeofanytype)foraspecifiedamountoftime(twominutesisusedinthisexperiment)inthehotwaterbathandanalyzethetemperaturechangeofthemetalrodovertime.ThetemperatureatthewaterandthefreeendofthemetalrodismeasuredwithdigitalthermometersaccuratetoonedecimalplaceandisnotedusinganExcelsheet.Holesweremachinedoutoftheendsofthealuminumrodsbyalocalmachineshop(CADdrawingsattached)andareofequivalentdepthanddiameterinalloftherods(theyweremachinedtoholdathermometerthatwasusedintheexperiment).Therodsaresubjectedtothe70°Cbathforexactly2.00minutesinalloftheexperimentsinordertoensurethatexactlythesameamountofheatenergyistransferredtotherodsineachoftheexperiments.ThedataisenteredintoaC#programthatisspecificallywrittenfortheexperimentand“q”theheattransferrateforthemetalisdeterminedforarodofknownlength.Theequationusedisshownbelow:
WhereA=cross‐sectionalarea,k=thethermalconductivity(204.8W/(m*K)foraluminum),x2‐x1=thelengthoftherod,andT1‐T2=thetemperaturesoftheimmersedendsubtractedfromthefreeendrespectively.Aninsulatedaluminumrodofunknownlengthisthensubjectedthesamehotwaterbathandthesameconstraints(i.e.heldinthebathatthesamedepth,time,etc.asintheinitialrun).ItstemperatureatthewaterendandfreeendareenteredintoanExcelsheetandthenintotheC#program.Thelengthoftherodisinstantlycalculatedusingtheqcalculatedfromtherodofknownlength.Thecalculatedlengthoftheunknownrodisthencomparedwithitsactuallengthusingaruler.Theexpectedoutcomeistohavethecalculatedlengthofthealuminumrodbeequivalenttoitsactuallength.
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REQUIREDMATERIALS
Table1:RequiredMaterialsItem Quantity
AluminumRod(Length=4in,Diameter=3/16in) 1AluminumRod(UnknownLength,Diameter=3/16in) 2
CylindricalStyrofoamPiece(Insulation) 3HotPlate(MaintainconstantTforwaterbath) 1Thermometers(Preferably1digital/1alcohol) 2
250mLBeaker 1Stand 1Clamp 2‐3
Stopwatch 1MagneticStirBar 1
Water 500mL
EXPERIMENTEQUATIONSOurexperimentofdeterminingthelengthofarodbasedontheamountofheattransferanddifferent thermal conductivity for differentmaterials canbe calculatednumerically using thefollowingequation:
From Christie J. Geankoplis’ Transport Processes and SeparationProcess Principle Chapter 4.2 Conduction Heat Transfer, thefollowingequationcanbeusedtodeterminethelengthoftherod:
Derivedinto:
Where q is the heat‐transfer rate in Watts (W), k is the thermalconductivity in W/(m∙K), A is the cross sectional area of the rod(π*D2/4m2), x is the length of the rod inmeter (m), and T is thetemperatureoftherodattwoendpoint(K).Todetermineq,usearodofknownlengthandsetitintothesamebathatthesametemperaturefor2minutes,usedthekvaluegiven(206(W/m∙K)),thetemperaturerecordedafter2minutes,areawithknowndiameterandlength.
T2
T1
D
x
1
2
Rod
Figure3:MetalRodDiagram
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EXPERIMENTALOUTCOMETheresultsgatheredbytheexperimentarewithinaexceptionalstandarddeviationtoclaimthatthereisahighaccuracythatcanbeachievedwiththisprocedure.Over100experimentswereperformedusingseveraldifferenttypesofrods(brass,stainlesssteelandaluminum)ofdifferentdiameters.AnExcelsheetisgeneratedthatgivesanaverageandstandarddeviationforthealuminumrodsusingexperimentsfromahotpotfilledwithfilteredwaterandmaintainedatapproximatelyat70°Cathome.AtablegeneratedbyExcelisalsopresentedalongwithstatisticalsignificanceusingthestandarddeviationandthestudent’st‐testfortheconfidenceintervalcalculation.
Table2:RawData(Rod1)InitialTemp
RodInitialWater Time(min) FinalTempRod Difference Length
21.7 70.0 2.0 46.8 25.1 7.03
21.7 70.0 2.0 46.7 25.0 7.03
21.7 70.0 2.0 46.7 25.0 7.03
21.7 70.0 2.0 46.9 25.2 7.03
Table3:RawData(Rod2)InitialTemp
RodInitialWater Time(min) FinalTempRod Difference Length
21.6 70 2 39.6 18 Unkn.
21.8 70 2 39.8 18 Unkn.
21.7 70 2 39.6 17.9 Unkn.
21.7 70 2 39.6 17.9 Unkn.
Table4:StatisticalData Rod1(FourTrials) Rod2(FourTrials)
Average 46.8°C 39.65°CStandardDeviation 0.096°C 0.1°C
ConfidenceInterval(90%) 46.8±0.112944 39.65±0.117655
Thedatafromthestudent’stconfidenceintervaltestrevealsthattheexperimentisstatisticallysignificantwithintherangesprovidedabove.Thecalculatedlengthoftherodis9.18cmandtheactuallengthoftherodis9.20cm.Hence,theexperimentisverysensitiveifitisperformedcarefullyenough.Becauseofthestatisticalvalidation,ageneralconclusioncanbemadeabouttheexperiment.Accuraterodmeasurementscanbedeterminedifcarefulattentionto1)bathtemperature2)insulationlengthandcoverage3)equipmentused(fast
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responsedigitalthermometers)and4)equivalenttimestherodsareimmersedinthewaterbath,aremaintained.CONCLUSIONTheequationusedinthisexperimentimpliesthatsteady‐stateconditionsareemployed.Inordertosatisfytherepetitionandtimerequirementconditionsfortheexperiment,twominutesareusedforeachtrial(toensurethesameamountofheattransfertoeachoftherods),hencetheyarenotinsteady‐state.Ideally,thisexperimentshouldbecarriedoutinamannerwheretwoindividualinsulatedmetalrodsofdifferentlengthareimmersedinaconstant‐temperaturewaterbathuntiltheyreachsteadystate.Thenthesameequationcanbeusedtocalculatetheheatlossforthemetalrodsthatareinparallelwiththeinsulation.Aweaknessoftheexperimentisthelackofhighqualitylabequipmenttotesttheequationsfromtheheattransferportionoftheclass.AhotpotthatwaspurchasedyearsagofromWal‐martisusedalongwithtwodigitalthermometersofunknownaccuracy.Theresultsprovedgoodinthisexperimentforrodsthatdifferedinlengthofeachotherbyapproximately2.0centimeters,butledustowonderwhetherthesameexperimentwouldgiveasgoodofresultsforrodsdifferingasmuchas10.0cm,forexample.Totalmaterialcostsinthisexperimentwereapproximately$15forthealuminumrods,machiningcosts,anddigitalthermometer.Futureusesofthisexperimentcanhavebroadimplications.Forexample,ifaplumberwantedtomeasureacertainlengthofpipethroughafloorwithoutcuttingthroughthefloorshe/hecouldheatthetopofthepipetoaknowntemperature,letthepipereachsteadystate,andmeasurethetemperatureofthepipebelow.Certainly,aroughestimateofthepipecouldbeobtainedinthismanner.
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APPENDIXA
A‐1:CADDrawingofMachinedRod