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