1: Heat conduction through a large plane wall Steady state heat transfer through pipes is in the normal direction to the wall surface (no
If the wall is hot, and we consider a liquid being heated in the pipe, the viscosity of the wall fluid will be smaller than that of the fluid in the bulk This
Heat transfer through a wall is one- dimensional when the temperature of the wall varies in one direction only FIGURE 3–2 Under steady conditions, the
The heat transfer from the fluid to the pipe walls is estimated and the temperature fluid and wall temperatures may vary along the length of the tube
and air sides and metal wall calculate the heat loss from 100 m length of the pipe Also find temperature drop across the insulation
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127898_3StaedyConductionHeatTransfer.pdf M.BahramiENSC388(F09)SteadyConductionHeatTransfer1
SteadyHeatConduction
Inthermodynamics,weconsideredtheamountofheattransferasasystemundergoesa processfromoneequilibriumstatetoanother.Thermodynamicsgivesnoindicationof howlongtheprocesstakes.Inheattransfer,wearemoreconcernedabouttherateof heattransfer.
Thebasicrequirementforheattransferis
thepresenceofatemperaturedifference.The temperaturedifferenceisthedrivingforceforheattransfer,justasvoltagedifferencefor electricalcurrent.ThetotalamountofheattransferQduringatimeintervalcanbe determined from:kJdtQQ t 0 Therateofheattransferperunitareaiscalledheatflux,andtheaverageheatfluxona surfaceisexpressedas 2 /mWAQq
SteadyHeatConductioninPlaneWallsConductionisthetransferofenergyfromthemoreenergeticparticlesofasubstanceto
theadjacentlessenergeticonesasresultofinteractionsbetweentheparticles.
Considersteadyconduction
throughalargeplanewallofthicknessȴx=Landsurfacearea
A.ThetemperaturedifferenceacrossthewallisȴT=T
2 -T 1 . Notethatheattransferistheonlyenergyinteraction;theenergybalanceforthewallcan beexpressed:dtdEQQ wall outin
ForsteadyͲstateoperation,
.constQQ outin Ithasbeenexperimentallyobservedthattherateofheatconductionthroughalayeris proportionaltothetemperaturedifferenceacrossthelayerandtheheattransferarea, butitisinverselyproportionaltothethicknessofthelayer.WxTkAQCond thickness)difference eraturearea)(temp (surfacefer heat trans of rate M.BahramiENSC388(F09)SteadyConductionHeatTransfer2
Fig.1:Heatconductionthroughalargeplanewall.
Theconstantproportionalitykisthethermalconductivityofthematerial.Inthelimiting casewhereȴxї0,theequationabovereducestothedifferentialform:
WdxdTkAQCond
whichiscalledFourier'slawofheatconduction.ThetermdT/dxiscalledthetemperature gradient,whichistheslopeofthetemperaturecurve(therateofchangeoftemperature
Twithlengthx).
ThermalConductivity
Thermalconductivityk[W/mK]isameasureofamaterial'sabilitytoconductheat.The thermalconductivityisdefinedastherateofheattransferthroughaunitthicknessof materialperunitareaperunittemperaturedifference. Thermalconductivitychangeswithtemperatureandisdeterminedthroughexperiments.
Thethermal
conductivityofcertainmaterialsshowadramaticchangeattemperatures nearabsolutezero,whenthesesolidsbecomesuperconductors. Anisotropicmaterialisamaterialthathasuniformpropertiesinalldirections. Insulatorsarematerialsusedprimarilytoprovideresistancetoheatflow.Theyhavelow thermalconductivity. A A
ȴxQ
• T 1 T 2 M.BahramiENSC388(F09)SteadyConductionHeatTransfer3
TheThermalResistanceConcept
TheFourierequation,forsteadyconductionthroughaconstantareaplanewall,canbe written:
LTTkAdxdTkAQ
Cond21
ThiscanbereͲarrangedas:
)/()( 12
WCkALRW
RTTQ wallwall Cond x R wall isthethermalresistanceofthewallagainstheatconductionorsimplytheconduction resistanceofthewall. Theheattransferacrossthefluid/solidinterfaceisbasedonNewton'slawofcooling: )/(1WChARWTThAQ Convs R conv isthethermalresistanceofthesurfaceagainstheatconvectionorsimplythe convectionresistanceofthesurface.
ThermalradiationbetweenasurfaceofareaAatT
s andthesurroundingsatT ь canbe expressedas: