Thermodynamics and Heat Transfer Reviewver. 1ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 20091Areas of Interest:Areas of Interest:Mechanical Engineering• Mechanics•ThermodynamicsThermodynamics• Heat TransferFl id•FluidsME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 20092Heat Transfer MechanismsHeat Transfer Mechanisms•ConductionConduction• Convectionto a limited extent via h–to a limited extent via h• Radiation–eb= σT4ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 200931D Conduction Heat Transfer2TT∂∂y2xTtT∂∂=∂∂αT = temperaturext = timeα = thermal diffusivity2lME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 20094Thermal Diffusivity (α)y()()tyconductivik()()( )heatspecificdensitytyconductivi∗=∗=ckραlh2[]timelength2=ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 20095Characteristic Time Constant (τ)()yτ2~lατxlh titidi i2ll = characteristic dimensionME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 200961D Lumpedcapacity HT (1)1D, Lumped-capacity HT (1)Bodies without temperature gradients:Biot number (Bi)= hl/k << 1h = heat transfer coefficienty∞Tsemiinfinite plateJoseph Fourier1768-1830x2liTsemi-infinite plate176818302liTJeanBaptiste BiotME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 20097Jean-Baptiste Biot1777-18621D, Lumped-capacity HT (2)1D, Lumpedcapacity HT (2)•First Law formulation of lumped-First Law formulation of lumpedcapacity model (pure thermal system):system):Heat transfer =Increase in Internal Heat ContentQ1221UUQ−=−ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 20098First Law Formulation()cdTVdtTTAhdQρ=∗−∗=∞•t = time()Qρ∞• T = temperature (∞= source or sink)•V = volumeA•A = area• h = overall heat transfer coefficient•ρ= density•ρ= density• c = specific heatME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 20099Heat Transfer TimeSolving for time (t)g()⎟⎞⎜⎛−∞1lTTcVρ⎟⎟⎠⎜⎜⎝−=∞∞21lnTTTTAhcVtρNote: k drops out because of assumption of negligible temperat re gradientsnegligible temperature gradients.ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 2009101D Heat Transfer-Ex. 1-11D Heat Transfer Ex. 11•You are interested in cooling a 0.5You are interested in cooling a 0.5 in. thick plate of copper from T=500oC to T=50oC.•It is cooled such that the heatIt is cooled such that the heat transfer coefficient (h) is 200 W/m2*K, and the “sink” temperature is 20oC.ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 2009111D Heat Transfer-Ex 1-21D Heat Transfer Ex. 12How long will it take to cool the plate down?down?ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 2009121-D Heat Transfer-Ex. 1-31D Heat Transfer Ex. 13Assume the plate can be modeled ifiit ltas an infinite plate. – Note: V/A = l (if l<< plate width and lth)b lit tdlength), because we are only interested in half of the plate, by symmetry•Copper data:•Copper data:– density (ρ) = 8,970 kg/m3d ti it (k) 393 W/ *K–conductivity (k) = 393 W/m*K– specific heat (c) = 385 J/kg*KME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 2009131D Heat TransferEx 141-D Heat Transfer -Ex. 1-40.00635*200hl#Biot==10 00323393k<<=10.00323<<=ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 2009141-D Heat Transfer - Ex. 1-5⎟⎞⎜⎛−∞1TTcVρ⎟⎟⎠⎞⎜⎜⎝⎛−=∞∞21lnTTTTAhcVtρ50020ln385 8,97000635.0⎟⎠⎞⎜⎝⎛−∗∗∗=tmins5s3045020200≈=⎟⎠⎜⎝−ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 200915Bodies with Temperature GradientsBodies with Temperature Gradients•Biot number (Bi)=hl/k>>1Biot number (Bi) hl/k 1– More difficult, use Heisler charts⎟⎟⎠⎞⎜⎜⎝⎛∂∂=∂∂22TtTαy⎟⎠⎜⎝∂∂xtxθBi-12lFoME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 2009162lFoSummarySummary•Reviewed 1D lumped-capacityReviewed 1D, lumpedcapacity conduction heat transfer•Main mechanism in this class•Main mechanism in this classME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT 200917ME 6222: Manufacturing Processes and Systems Prof. J.S. Colton © GIT
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