Heat Chap03064.docx
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Heat Chap03064.docx
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HeatChap03064
HeatConductioninCylindersandSpheres
3-64CWhenthediameterofcylinderisverysmallcomparedtoitslength,itcanbetreatedasanindefinitelylongcylinder.Cylindricalrodscanalsobetreatedasbeinginfinitelylongwhendealingwithheattransferatlocationsfarfromthetoporbottomsurfaces.However,itisnotpropertousethismodelwhenfindingtemperaturesnearthebottomandthetopofthecylinder.
3-65CHeattransferinthisshortcylinderisone-dimensionalsincetherewillbenoheattransferintheaxialandtangentialdirections.
3-66CNo.Insteady-operationthetemperatureofasolidcylinderorspheredoesnotchangeinradialdirection(unlessthereisheatgeneration).
3-67Asphericalcontainerfilledwithicedwaterissubjectedtoconvectionandradiationheattransferatitsoutersurface.Therateofheattransferandtheamountoficethatmeltsperdayaretobedetermined.
Assumptions1Heattransferissteadysincethespecifiedthermalconditionsattheboundariesdonotchangewithtime.2Heattransferisone-dimensionalsincethereisthermalsymmetryaboutthemidpoint.3Thermalconductivityisconstant.
PropertiesThethermalconductivityofsteelisgiventobek=15W/m°C.Theheatoffusionofwaterat1atmis
.Theoutersurfaceofthetankisblackandthusitsemissivityis=1.
Analysis(a)Theinnerandtheoutersurfaceareasofsphereare
WeassumetheoutersurfacetemperatureT2tobe5°Caftercomparingconvectionheattransfercoefficientsattheinnerandtheoutersurfacesofthetank.Withthisassumption,theradiationheattransfercoefficientcanbedeterminedfrom
Theindividualthermalresistancesare
Thenthesteadyrateofheattransfertotheicedwaterbecomes
(b)Thetotalamountofheattransferduringa24-hourperiodandtheamountoficethatwillmeltduringthisperiodare
Check:
Theoutersurfacetemperatureofthetankis
whichisveryclosetotheassumedtemperatureof5Cfortheoutersurfacetemperatureusedintheevaluationoftheradiationheattransfercoefficient.Therefore,thereisnoneedtorepeatthecalculations.
3-68Asteampipecoveredwith3-cmthickglasswoolinsulationissubjectedtoconvectiononitssurfaces.Therateofheattransferperunitlengthandthetemperaturedropsacrossthepipeandtheinsulationaretobedetermined.
Assumptions1Heattransferissteadysincethereisnoindicationofanychangewithtime.2Heattransferisone-dimensionalsincethereisthermalsymmetryaboutthecenterlineandnovariationintheaxialdirection.3Thermalconductivitiesareconstant.4Thethermalcontactresistanceattheinterfaceisnegligible.
PropertiesThethermalconductivitiesaregiventobek=15W/m°Cforsteelandk=0.038W/m°Cforglasswoolinsulation
AnalysisTheinnerandtheoutersurfaceareasoftheinsulatedpipeperunitlengthare
Theindividualthermalresistancesare
Thenthesteadyrateofheatlossfromthesteamperm.pipelengthbecomes
Thetemperaturedropsacrossthepipeandtheinsulationare
3-69
"GIVEN"
T_infinity_1=320"[C]"
T_infinity_2=5"[C]"
k_steel=15"[W/m-C]"
D_i=0.05"[m]"
D_o=0.055"[m]"
r_1=D_i/2
r_2=D_o/2
"t_ins=3[cm],parametertobevaried"
k_ins=0.038"[W/m-C]"
h_o=15"[W/m^2-C]"
h_i=80"[W/m^2-C]"
L=1"[m]"
"ANALYSIS"
A_i=pi*D_i*L
A_o=pi*(D_o+2*t_ins*Convert(cm,m))*L
R_conv_i=1/(h_i*A_i)
R_pipe=ln(r_2/r_1)/(2*pi*k_steel*L)
R_ins=ln(r_3/r_2)/(2*pi*k_ins*L)
r_3=r_2+t_ins*Convert(cm,m)"t_insisincm"
R_conv_o=1/(h_o*A_o)
R_total=R_conv_i+R_pipe+R_ins+R_conv_o
Q_dot=(T_infinity_1-T_infinity_2)/R_total
DELTAT_pipe=Q_dot*R_pipe
DELTAT_ins=Q_dot*R_ins
Tins[cm]
Q[W]
Tins[C]
1
189.5
246.1
2
121.5
278.1
3
93.91
290.1
4
78.78
296.3
5
69.13
300
6
62.38
302.4
7
57.37
304.1
8
53.49
305.4
9
50.37
306.4
10
47.81
307.2
3-70A50-mlongsectionofasteampipepassesthroughanopenspaceat15C.Therateofheatlossfromthesteampipe,theannualcostofthisheatloss,andthethicknessoffiberglassinsulationneededtosave90percentoftheheatlostaretobedetermined.
Assumptions1Heattransferissteadysincethereisnoindicationofanychangewithtime.2Heattransferisone-dimensionalsincethereisthermalsymmetryaboutthecenterlineandnovariationintheaxialdirection.3Thermalconductivityisconstant.4Thethermalcontactresistanceattheinterfaceisnegligible.5Thepipetemperatureremainsconstantatabout150Cwithorwithoutinsulation.6Thecombinedheattransfercoefficientontheoutersurfaceremainsconstantevenafterthepipeisinsulated.
PropertiesThethermalconductivityoffiberglassinsulationisgiventobek=0.035W/m°C.
Analysis(a)Therateofheatlossfromthesteampipeis
(b)Theamountofheatlossper
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- Heat Chap03064