GENERAL ASPECTS OF HEAT TRANSFER& 1-DIMENSIONAL STEADY-STATE CONDUCTION Diference Between Temperature and HeatTemperatureisameasureoftheamountofenergy storedbythemoleculesofasubstance.Itmanifests itself as a degree of hotness, and can be used topredictthedirectionofheattransfer.Thedrivingforceforanyformofheattransferisthe temperature diference.Heat,isenergyintransit.Heatl i keworki sa transi entquanti tywhi chonl yappearsatthe boundary(hi l eachangei staki ngpl acewi thi n thesystem!"Temperaturehasonlymagnitude.itisascalar quantity.Heattransferhasdirectionaswellas magnitude. it is a vector quantity.Thermodynamicstellsus: how much heat is transferred(Q)how much work is done (W)final state of the systemHeat transfer tells us: how (with what modes)Q is transferred at what rate Q is transferredtemperature distribution inside the bodyDifference b/w Thermodynamics and HeatTransferMechanisms(Modes) of Heat Transfer Conduction:transfer of heat across asubstancewithoutappreciable displacementofmolecules forming the substanceConvection:heattransferbyfluid currents! in a gas or li"uid# due natural floworforcedflow$%tisanimportant mechanismofenergytransferbetween a solid surface and a li"uid or a gas$&adiation:heattransferdueto electromagneticwavesacrossa space# even in a vacuum$ 'll mechanisms may be involved in practical heat transfer#but the dominant mechanism differs in different cases$Heat #ow depends on the temperature difference. Heat ConductionConduction is the transfer of heat through materials by the direct contact of matter$(ensemetalslikecopperandaluminumareverygoodthermal conductors$' thermal insulator is a material that conducts heat poorly$ConvectionConvectionisthetransferofheat by the motion of li"uids and gases$Convection in a gas occurs because gas e)pands when heated$Convectioninli"uidsoccurs because of differences in density$Hotwaterislikewiselessdensethancoldwaterandrises# causing convection currents which transport energy$ Radiation&adiationisheattransferby electromagnetic waves$Thermalradiationis electromagneticwaves(including light)producedbyob*ectsbecause of their temperature$Thehigherthetemperatureofan ob*ect#themorethermalradiation it gives off$ +ower absorbed by surface of an ob*ect,absabs sdQP Aa Tdt A, aQabsTs-urface 'rea# A'bsorption coefficient# a . / to 01missivity# e . / to 0-tefan23olt4mann constant 5 . 6$78 ) 0/29 W$m2:$;2, +ower radiated from the surface of an ob*ect,radraddQP Ae Tdt A, e, T Qradnet rad abs+ P P Pnet < / net heat transfer out of systemFouriers !aw of Heat ConductionTherateofheatconductionthroughamediuminaspecified direction (say# in the x2direction) is e)pressed by =ourier>s law of heat conduction for one2dimensional heat conduction as: Heatisconductedinthedirectionof decreasingtemperature#andthusthe temperaturegradient(dT?d@)isnegative whenheatisconductedinthepositivex2direction$

dxdTkA Q =orconductionbetweentwoplanesurfaces(egheatloss through the wall of a house) the rate of heat transfer isH CQ T Tk At L dQ dTk Adt dx Thermal conductivity k (W$m20$;20)steady2stateheat currentH . dQ?dtenergy transferred through slabQTHT$LQQAH CQ T Tk At L dQ dTk Adt dx steady2state Thermal Conduction through a uniform slab0xLT$THtemperature gradientdTdxHeat F"u# Heat flu) is the flow of heatper unit area and per unit time $ %t is directly proportional to the temperature gradient$=rom one dimensional =ourierAs law:where:" . the heat flu)k . the coefficient of thermal conductivityT . the temperature) . a spatial coordinate

dxdTk qAQ #dxdTkA Q Therma" ConductivityThe thermal conductivity of a material describes how well the material conducts heat$$onductivitydependsonhowmoleculesare structurallybondedtogether,whichrelatestothe state of matter. -olids Bolecules strongly bonded High conductivityCi"uids Bolecules weakly bonded -ome conductivityDases Bolecules not bonded Cow conductivityBaterial Thermal conductivity k (W$m20$;20)Diamond2450 Cu385 Al205Glass0.8Water0.6Wood0.2Foam0.01Air0.024 Thermal conductivity (k ) is theproperty of the material Thermal conductivity ofmaterialisdueto #owoffreeelectrons (inmetals!andlattice vibration ( in #uids! % (&ure metal!'

%(alloy!'%((iq.!'%(gas! $"ectrica" %na"o&y of Heat Transfer)resistancecanbede*nedastheratioofa driving potential to a corresponding transfer rate.iVR1lectricalresistanceistoconductionofelectricityasthermal resistance is to conduction of heat$TheanalogofQiscurrent#andtheanalogofthetemperature difference# T1 - T2, is voltage difference$ =rom this perspective the heat transfer from aslab is written as: totalRT TQ: 0 Therma" Resistance For Conduct'ondxdTkA Qwall cond #(!Temperaturethroughthewallvaries linearlywith@$Temperature distributioninthewallundersteady conditions is a straight line$ :0#/TT Twall condLxkAdT dx QLT TkA Qwall cond: 0#wallwall condRT TQ: 0#(!kALRwall(+$ , !Thermal&esistance dependsonthegeometry andthethermalproperties of the mediumeR V VI: 0 ALReeeR. 1lectrical resistance: 0V V - .oltage diference across the resistancee-/lectrical conductivityLT TkA Qwall cond: 0#Therma" Resistance For Convect'onconvSconvRT TQconvR. Convection resistance of surface) ( T T A QS S conv(!SconvAR0 (+$ , !

- Convection heat transfer coefficientTherma" Resistance For Radiation!ads"!! Ss"!! S S !ad !adR T TT T AQ ) (S !ad!adA R0) (s"!! S S!ad!adT T AQ

!ad conv co#$ined + ) (, ,s"!! S S !adT T A Q Thermal &esistance -eries 'rrangementtotalRT TQ: 0 AA kLA kLA RR R R R Rtotalconv wall wall conv total: ::000: # : # 0 # 0 #0 0+ + + + + + Thermal &esEstance +arallel'rrangement)0 0)( (: 0: 0:: 00: 0: 0R RT TR T TR T TQ Q Q + + + totalRT TQ: 0 : 00 0 0R R Rtotal+ : 0: 0R RR RRtotal+ $omb0ned 1er0es2&arallel )rrangementtotalRT TQ0conv conv totalR RR RR RR R R R + ++ + + F: 0: 0F 0:0 000A kLR: :::A kLR F FFFA kLR F0ARconv ()$ D*M$)+*()%! +T$%D, +T%T$ H$%T C()D-CT*()Consideraplanewallbetween twofluidsofdifferent temperatureThe ."ane /a""Heat 1"uation:/d dTkdx dx _ ,Temperature (istribution for Constant k:( )( )#0 #: #0 s s sxT x T T TL + Heat =lu) and Heat &ate:( )#0 #: x s sdT kq k T Tdx L ( )#0 #: x s sdT kAq kA T Tdx L Thermal &esistances:Conduction in a plane wall:# t condLRkAConvection: #0t convRAThermal &esistance for Gnit -urface 'rea:# t condLRk#0t convR

&adiation#0t !ad!R A#0t !ad!R

Thermal circuit for plane wall with ad*oining fluids:0 :0 0totLRA kAA + +#0 #:xtotT TqR Composite Wall with Hegligible Contact &esistance:#0 #,xtotT TqR 0 ,0 0 0C tot A %totA % CL R L LRAk k kA 1 + + + + 1 ]Iverall Heat Transfer Coefficient (&) :'modifiedformofHewton>sCawofCoolingtoencompass multiple resistancesto heat transfer$x ove!allq &AT 0totR&ATherma" Contact Resistance'a( contactQ Q Q + e!)ace cT AQint e!)acecTA Q

int?(,m3 +$!(m3 +$, !A QT

Re!)acecc?0int h$4 thermal contact conductance Thermal contact resistance is inverse of thermal contact conductionJaluesdependon:Baterialsoflayer0and:#surfacefinishes# and contact pressure Heatislostfroma hot2waterpipetothe air outside in the radial direction$Heattransferfroma longpipeisone dimensionalHeat 1"uation:0/d dTk!! d! d! _ ,Temperature (istribution for Constant k( )( )#0 #:#:0 : :lnln ?s ssT T!T ! T! ! ! _ + , Heat Conduct'on in Cy"'nderHeat =lu) and Heat &ate:( )( )( )( )( )( )#0 #:: 0#0 #:: 0#0 #:: 0ln ?::ln ?::ln ?! s s! ! s s! ! s sdT kq k T Td! ! ! !kq !q T T! !Lkq !Lq T T! ! Conduction &esistance4( )( ): 0#: 0#ln ?Gnits ;?W:ln ?Gnits m ;?W:t condt cond! !RLk! !Rk Composite TubeWall :( )#0 #,#0 #, !totT Tq &AT TR ( )0i i tot& A RHeat 1"uation: +pherica" +he""::0/d dT!d! d!! _ ,Temperature (istribution for Constant( )( )( )( )0?#0 #0 #:0 :00 ?s s s! !T ! T T T! ! Heat flu)# Heat &ate and Thermal &esistance:( ) ( )( )#0 #::0 :0? 0?! s sdT kq k T Td!! ! ! 1 ]( ) ( )( ):#0 #:0 :,,0? 0?! ! s skq ! q T T! ! ( ) ( )0 :#0? 0?,t cond! !Rk Composite -hell:ove!all! ove!alltotTq &ATR 0Constanttot&A R ( )0 (epends on i i tot i& A R A Cr't'ca" Rad'us of *nsu"at'on) : (0:) ? ln(:0 :0 0L !Lk! !T TR RT TQconv ins ++ / ?: d! Q d

k!c*linde! c!#k .Thermal conductivityh . 1)ternal convection heat transfer coefficient

k!s(e!e c!:#Thickness of insulation up to which heat flow increases and after which heat flowdecreases$ -election of %nsulatEon ThEcknessc!c!c!! !! !! !>