Heat Transfer Conduction And Convection
Steady Heat TransferFebruary 14, 2007OutlineSteady Heat Transfer withConduction and Convection Review last lecture Equivalent circuit analysesLarry CarettoMechanical Engineering 375– Review basic concept– Application to series circuits withconduction and convection– Application to composite materials– Application to other geometriesHeat TransferFebruary 14, 2007 Two-dimensional shape factors2Review Steady, 1-D, e& gen 0 Rectangulark (TL T0 )Q&q& ALReview Heat Generation Cylindrical shellQ&2πk (T2 T1 ) Lln (r2 r1 )Figure 2-21 from Çengel, VariousHeat and Mass Transferphenomena insolids cangenerate heat Define e& genas the heatI 2 ρLgenerated perI 2ρI 2R A 2unit volume e&gen LAVAper unit time Spherical shell k is an average thermal4πk (T2 T1 )conductivity (or a constantQ& value) if k is constant1 / r1 1 / r2T0, TL temperatures at x 0,L; T1, T2 temperatures at inner (r1) and outer(r2) radii34Plot of (T - T0)/(TL - T0) for Heat Generation in a SlabReview Heat Generation II21.8 Temperature and heat flux equations2k e&genxL2kTemperatureDifference RatioT T0 e&genx21.6(T T )x 0 LLe&gen(2x L) k (T0 TL ) q& 2L1.4H 01.2H .01H .1H 11H 2H 50.8H 100.62H 0.40.2Q& x 0 E& gen Q& x LL e&genk (TL T0 )000.10.20.30.40.50.60.70.80.91x/L5ME 375 – Heat Transfer61
Steady Heat TransferFebruary 14, 2007Steady Heat Transfer DefinitionRectangular Steady ConductionFigure 3-2 fromÇengel, Heatand MassTransfer In steady heat transfer the temperatureand heat flux at any coordinate point donot change with time Both temperature and heat transfer canchange with spatial locations, but notwith time Steady energy balance (first law ofthermodynamics) means that heat inplus heat generated equals heat outThe heattransfer isconstantin this 1Drectanglefor bothconstant& variablekdTQ& q& kAdx7Thermal ResistanceFigure 2-63 from Çengel,Heat and Mass Transfer8Thermal Resistance II Heat flow analogous to current Temperature difference analogous topotential difference Both follow Ohm’s law with appropriateresistance term Conductionk A(T1 T2 )T TQ& Q& 1 2LRcond Convection(Q& hA Ts T f) Q& Ts T fRconv Rcond LkA Rconv 1hA RadiationRrad (11A1F12σ T13 T23 T22T1 T12T2) Ah1 rad9Where Does the Heat Go?10Where Does the Heat Go? IIEnergy conservationrequires that conductionheat through wall equalsthe heat leaving the wallby convection andradiationFigure 1-18from Çengel,Heat andMassTransferQ&1 Q& 2 Q& 3Figure 1-18 fromÇengel, Heat andMass Transfer11ME 375 – Heat TransferFigure 3-5 from Çengel,Heat and Mass Transfer122
Steady Heat TransferFebruary 14, 2007Parallel Resistances (T Tsurr)111 Rtotal Rconv RradT 1Tsurr Rtotal As hconv As hradDefine total heat transfercoefficient, htotalFigure 3-5from Çengel,Heat andMass Transferhtotal 1 hconv hradAs RtotalCombined Modesq& h(T 1 T1 )Convection orconvectionplus radiationLQ& T 1 T 211L Ah1 kA Ah2 q& SeriesResistanceFormulaQ&T T 1 2A 1 L 1h1 k h2A is area normalto heat flow14Q&T T 1 21L 1A h1 k h2T1 T 1 ProblemA house has a 4 in thick brick wall with k 0.6Btu/hr·ft·oF. The interior temperature is 70oF andthe exterior temperature is 0oF. The inside andoutside convection plus radiation coefficients are3 Btu/hr·ft2·oF and 4 Btu/hr·ft2·oF, respectively.Find the heat flux through the wall.Given: Wall with L 4 in 4/12 ft and k 0.6Btu/hr·ft·oF has convection on two sides. T 1 70oF, T 2 0oF, h1 3 Btu/hr·ft2·oF and h2 4Btu/hr·ft2·oF.&QFind: q& 17AFigure 3-6 fromÇengel, Heat andMass TransferLIf you know h1, h2, L, k, T 1,and T 2, but you do not knowT1 and T2, can you find theheat flux?Once you found the heat flux from theinformation give, can you find T1 and T2?q& 15ME 375 – Heat Transferq& h(T2 T 2 )Combined Modes IIIFigure 3-6 fromÇengel, Heat andMass TransferT TT 1 T 2Q& 1 2 RtotalRconv,1 Rwall Rconv,1k (T1 T2 )LAll q& values are the sameCombined Modes IILq& Figure 3-6 fromÇengel, Heat andMass Transfer13A is area normalto heat flowConvection orconvectionplus radiationConductionq&h1T2 T 2 q&h216SolutionA is area normalto heat flowq& LFigure 3-6 fromÇengel, Heat andMass TransferQ&T T70o F 0o F 1 2 2oA 1 L 1hr ft F 4 hr ft o F hr ft 2 o F fth1 k h23 Btu120.6 Btu4 Btuq& Q& 61.5 Btu A hr ft 2Find values of T1 andT2. Can you checkthese values?183
Steady Heat TransferFebruary 14, 2007Solution IIA is area normalto heat flowT Tq& 1 11h1T Tq& 2 21h2Solution IIIFigure 3-6 fromÇengel, Heat andMass TransferLq& T1 T 1 70o F h1q& T2 T 2 0o F h161.5 Btuhr ft 2 49.5o F3 Btuhr ft 2 o F61.5 Btuhr ft 15.4o F4 Btu19hr ft 2 o FA is area normalto heat flowFigure 3-6 fromÇengel, Heat andMass TransferLHow can we check results below found from analysis ofoverall problem and convection processes?q& 61.5 Btuhr ft 2T1 49.5o FT2 15.4o FAnalyze conduction step for consistency.2q& 61.5 Btuhr ft 2Composite Materials(0.6 Btu49.5o F 15.4o FQ& k (T1 T2 ) hr ft o F 4ALft2012)Composite Materials IIHowwouldyouanalyzethisproblem?Figure 3-9 from Çengel, Heat andMass Transfer21Review Cylindrical ShellFor constant kQ& r 2πk (T1 T2 ) L r ln 2 r1 Figure 250 fromÇengel,Heat andMassTransferR Q& r ME 375 – Heat Transfer22Cylindrical Shell with ConvectionQ& A1 2πr1LRconv,1 r 1ln 2 2πkL r1 T1 T2T T 1 2R r 1ln 2 232πkL r1 T 1 T 2Rconv,1 Rcond Rconv, 2Rconv,2 A2 2πr2 LQ& 11 h1 A1 h1 2πr1L11 h2 A2 h2 2πr2 LT 1 T 2 r 111ln 2 h1 2πr1L 2πkL r1 h2 2πr2 LFigure 3-25 from Çengel, Heatand Mass Transfer244
Steady Heat TransferFebruary 14, 2007Cylinder plus Convection ResultQ& ProblemT 1 T 2 r 111 ln 2 h1 2πr1L 2πkL r1 h2 2πr2 L A hot-water pipe (k 35 Btu/hr·ft·oF) in ahouse, made of ¾ inch schedule 40 pipe(OD 1.050 in; ID 0.824 in) is 40 ftlong and contains water at 120oF. Theair around the pipe is at 60oF. The heattransfer coefficients inside and outsidethe pipe are, respectively, 200 and 3Btu/hr·ft2·oF. Determine the heat lossQ&2π(T 1 T 2 )from the pipe. We can rearrangethis equation asshown belowQ&2π(T 1 T 2 ) L1 1 r2 1 ln h1r1 k r1 h2r2Figure 3-25 from Çengel, Heatand Mass TransferL25Solution26Solution II60oF,Given: T 2 T 1 120oF, r1 ID/2 0.412 in,r2 OD/2 0.525 in, k 35Btu/hr·ft·oF, L 40 ft,h1 200Btu/hr·ft2·oF,h2 3 Btu/hr·ft2·oFFigure 3-25 fromÇengel, Heat andMass Transfer1 1 r2 1 ln h1r1 k r1 h2r2&Find: QQ&2π(T 1 T 2 ) L1 1 r2 1 ln h1r1 k r1 h227r2Given: T 2 60oF, T 1 11 12 inhr ft 2 o F 120oF, r1 ID/2 0.412 in,200 Btu 0.412 in fth1r1r2 OD/2 0.525 in, k 35Btu/hr·ft·oF, L 40 ft,1 r2 hr ft o F 0.525 in ln ln h1 200 Btu/hr·ft2·oF,k r1 35 Btu 0.412 in 2oh2 3 Btu/hr·ft · Fhr ft 2 o F11 12 in1,940 Btu Find: Q& hrh2r23 Btu 0.525 in ftQ& (CompositeCylindricalShell IICompositeCylindrical Shell)2π(T 1 T 2 )L2π 120oF 60oF (40 ft ) o1 1 r2 1 ln (0.146 0.007 7.619) hr ft Fh1r1 k r1 h2r2Btu 28Figure 3-26 fromÇengel, Heat andMass Transfer11 h2 A4 h2 2πr4 LFigure 3-26 from Çengel, Heatand Mass TransferME 375 – Heat Transfer291 r2 1 r3 ln ln 11 k1L r1 k2 L r2 h1 A1 h1 2πr1L1 r4 ln k3 L r3 305
Steady Heat TransferFebruary 14, 2007CompositeCylindricalShell IIIFigure 3-26from Çengel,Heat and MassTransferAnother Problem Insulation with k 0.2 Btu/hr·ft·oF is to beadded to the pipe in the previousexample problem. Determine the heattransfer if the insulation is one inch thick.Q& Q& T 1 T 2 r r2 r 11111 ln 4 ln 3 ln h1 2πr1L 2πk1L r1 2πk2 L r2 2πk3 L r3 h2 2πr4 LQ& T 1 T 2 r2 r 1111 ln ln 3 h1 2πr1L 2πk1L r1 2πk2 L r2 h2 2πr3 L2πL(T 1 T 2 )11 r2 1 r3 1 ln ln h1r1 k1 r1 k2 r2 h2r331Another Problem IIAnother Problem IIIQ& Unchanged resistances from previousexample 1 1 r2 0.153 hr ft o Fh1r1 ln k1 r1 Know all termsfrom previousexample except32these twoBtu New and modified resistances1 r3 hr ft o F 1.525 in 5.332 hr ft o F ln ln k2 r2 0.2 Btu 0.525 in Btu2πL(T 1 T 2 )11 r2 1 r3 1 ln ln h1r1 k1 r1 k2 r2 h2r3()2π 120oF 60oF (40 ft )(0.146 0.007 5.332 2.623) hr ft oF 1860 BtuhrBtu Insulation and outer convection resistancesare largesthr ft 2 o F11 12 in 2.623 hr ft o F h2r3Btu3 Btu 1.525 in ft– Inner convection and pipe conduction negligible– Outer convection resistance less with insulation3334Effect of Insulation ThicknessInsulation Increases Q& ?2500 Why does initial amount of insulationincrease heat transfer?20001500Heat loss(Btu/hr)10005000012345678910– Tradeoff of two resistances– Added insulation adds conductionresistance– Added insulation also increases outerradius which decreases the outerconvection resistance 1/(houterAouter) 1/(houter2πrouterL)Thickness (in)35ME 375 – Heat Transfer366
Steady Heat TransferFebruary 14, 2007Resistances for Pipe InsulationRadius for Maximum Q&1210Q& 8Resistance6(hr·ft·F/Btu)2000.511.52ro outer radiusko thermal conductivityof outer layer 1 11 2πL(T 1 T 2 ) k r h r2 dQ&2 o o o 0&:For maximum Q2dro ro 1 1 Rother ln ko ri h2ro stance42πL(T 1 T 2 )1 r 1Rother ln o ko ri h2 ro2.53&ro ko/h2 for maximum QInsulation Thickness (in)37Radius for Maximum Q&Spherical Shell with Convection& ro ko/h2 for maximum Q In the example problem h2 3Btu/hr·ft2·oF, and ko 0.2 Btu/hr·ft·oF so&ro 0.0667 ft 0.8 in for maximum Q Pipe radius was 0.525 in; ro 0.8 ingives an insulation thickness of 0.275 in Note that ro ko/h2 does not depend onri and is usually larger than ri& There is no radius for minimum Q39Spherical Shell ResultRsphQ& A2 4πr22 L4π(T 1 T 2 )1r r1 2 1 2kr1r2 h2 r22h1r1Figure 3-25 from Çengel, Heatand Mass TransferME 375 – Heat TransferQ& A1 4πr12 LRsphT 1 T 2Rconv,1 Rsph Rconv, 2Rconv,1 11 h1 A1 h1 4πr12Rconv,2 11 h2 A2 h2 4πr22A2 4πr22 LFigure 3-25 from Çengel, Heatand Mass TransferRsph1 1 r1 r2 4πk40Conduction Shape FactorsT 1 T 2Q& 1 1 11r1 r2 24kπh1 4πr1h2 4πr22A1 4πr12 L3841 Simplified analysis– for multidimensional geometries with eachsurface at a uniform temperature– Use shape factor, S, whose equation isfound from tables like Çengel Table 3-7& kS(T – T )– Basic equation: Q12– S must have dimensions of length Equations for S depend on parameters in thedifferent geometries427
Steady Heat TransferFebruary 14, 2007Buried Pipe Shape FactorExample Shape Factor3.53.0S/L2.52.01.51.00.50.00From Table 7-1 in Çengel, Heatand Mass TransferME 375 – Heat Transfer102030405060708090100z/D43448
Both temperature and heat transfer can change with spatial locations, but not with time Steady energy balance (first law of thermodynamics) means that heat in plus heat generated equals heat out 8 Rectangular Steady Conduction Figure 2-63 from Çengel, Heat and Mass Transfer Figure 3-2 from Çengel, Heat and Mass Transfer The heat .
Boiling water CONDUCTION CONVECTION RADIATION 43. Frying a pancake CONDUCTION CONVECTION RADIATION 44. Heat you feel from a hot stove CONDUCTION CONVECTION RADIATION 45. Moves as a wave CONDUCTION CONVECTION RADIATION 46. Occurs within fluids CONDUCTION CONVECTION RADIATION 47. Sun’s rays reaching Earth CONDUCTION CONVECTION RADIATION 48.
Heat Transfer: Conduction, Convection and Latent Heat In addition to radiation, energy can also be transferred in the form of heat. There are three ways this can happen: When heat simply diffuses through an object, passing from molecule to molecule, that's called conduction As it turns out, air is actually very poor at diffusing heat
describe how heat moves by conduction, convection, and radiation. give examples of heat transfers that occur in every day situations. Vocabulary conduction: the transfer of heat through a material by direct contact. convection: the transfer of heat in a fluid (gas or liquid) as a result of the movement of the fluid itself.
the voltage applied and resistance of the heat source respectively, then using the energy balance, the net heat transfer (Q net) within the enclosure is given by- QQ Q Qnet conduction convection radiation ( 2 net Q V R ) The conduction heat transfer (Q conduction) at the enclosure side walls is determined as- wall conduction wall wall wall .
Chapter 5 Principles of Convection heat transfer (Text: J. P. Holman, Heat Transfer, 10th ed., McGraw Hill, NY) 5-1 INTRODUCTION We now wish to examine the methods of calculating convection heat transfer and, in particular, the ways of predicting the value of the convection heat-transfer coefficient h. Our discussion in this chapter will
(ignoring external natural convection), conduction through the wall, and natural convection and radiation to the interior wall. An important factor in the selection of this project was that it included all three major heat transfer mechanisms: conduction, convection, and radiation. It also included both major types of convection: forced and .
convection, pure solutal convection, heat transfer driven flows, and mass transfer driven flows were taken into account. Mobedi et al.,8,9 analyzed double diffusive convection in partially heated cavities. Nikbakhti and Rahimi10 studied numerically the flow, heat and mass transfer in
Jeffery was a good introduction to scoping. In appropriate order different bureaucratic levels were tackled, always sensitive to the pressures in each place. The many discussions with Roger proved useful during the field work later. For example, we confronted the problem of finding very large sample sites which were suitable on other parameters. So we discussed how this should be tackled .
