Heat Transfer - CSUN

1 Final ReviewMay 16, 2006ME 375 heat Transfer1 Review for Final ExamReview for Final ExamLarry CarettoMechanical Engineering 375 heat TransferHeat TransferMay 16, 20072 Outline Basic equations, thermal resistance heat sources Conduction, steady and unsteady Computing convection heat Transfer Forced convection, internal and external Natural convection Radiation properties Radiative Exchange3 Final Exam Wednesday, May 23, 3 5 pm Open textbook/one-page equation sheet Problems like homework, midterm and quiz problems Cumulative with emphasis on second half of course Complete basic approach to all problems rather than finishing details of algebra or arithmeticBasic Equations Fourier law for heat conduction (1D)()()LTTkAAqQorLTTkq2121 == =&&& Convection heat Transfer )( =TThAQssconv& Radiation (from small object, 1, in large enclosure, 2)()

2 42411121,TTAQrad = &5 heat Generation Various phenomena in solids can generate heat Define as the heat generated per unit volume per unit timegene&Figure 2-21 from engel, heat and Mass Transfer6 rectangular Energy BalancegenzyxpezqyqxqtTc&&&&+ = heat generatedStored energyheat inflow heat outflowgenpezTkzyTkyxTkxtTc&+ + + = +=Uses Fourier Law = Tkq&Final ReviewMay 16, 2006ME 375 heat Transfer27 Cylindrical CoordinatesgenpezTkzTkrrTkrrrtTc&+ + + = 211dzrdrTkdAqQdrr ==&&Figure 2-3 from engel, heat and Mass Transfer8 Spherical CoordinatesgenpeTkrTkrrTkrrrtTc&+ + + = 22222sin1sinsin11 Figure 2-3 from engel, heat and Mass Transfer91-D, rectangular , heat Generation Temperature profile for generation with T = T0at x = 0 and T = TLat x = L() + = =LTTkLekxekdxdTkqLgengen0222&&&()()LTTkL xeqLgen + =022&&()LxTTkxLekxeTTLgengen + =02022&&10 Plot of (T - T0)/(TL - T0) for heat Generation in a 1x/LTemperatureDifference RatioH = 0H =.

3 01H = .1H = 1H = 2H = 5H = 1010()02 TTkeLHLgen =&11 Slab With heat GenerationBoth boundary temperatures = Distance, x/LT / TB H = 0H = .01H = .1H = 1H = 2H = 5H = 10 BgenkTeLH&2=12 Thermal Resistance Conduction()kALRRTTQLTTAkQcondcond= = =2121&& Convection()hARRTTQTThAQconvconvfsfs1= = =&& Radiation()radradhATTTTTTAR1221122323112 111=+++ =FFinal ReviewMay 16, 2006ME 375 heat Transfer313 Composite Materials II14 LrhAh1111211 =LrhAh4242211 = 121ln1rrLk 232ln1rrLk 343ln1rrLkComposite Cylindrical ShellFigure 3-26 from engel, heat and Mass Transfer15 Fin Results Infinitely long fin heat Transfer at end (Lc= A/p)()mLxLmTTmLxLmTTcbccbcosh)(coshcosh) (cosh = = = ()

4 MLTThpkAQbcxtanh0 = =&()ckAhpxbmxbeTTTTe = = () == ==TThpkAqAQbcxcx00&&16 Fin Efficiency Compare actual heat Transfer to ideal case where entire fin is at base temperature() == TThAQQQ bfinfinfinfinfin&&&max,pLAcfin=for uniform cross sectionFigure 3-39 from engel, heat Transfer17 Overall Fin Effectiveness Original area, A = (area with fins, Afin) + (area without fins, Aunfin)()()() +=TThATTAAhQQbfinnobunfinfinfinfinnofin &&Figure 3-45 from engel, heat Transfer +==finnounfinfinnofinfinfinnofintotalAAA AQQ &&18 Lumped Parameter ModelcppLchchAb ==V Assumes same temperature in solid Use characteristic length Lc= V/A()()() + = = TeTTToreTTTT btibti Must have Bi = hLc/k < to use thisFinal ReviewMay 16, 2006ME 375 heat Transfer419 Transient 1D ConvectionFigure 4-11 in engel, heat and Mass TransferAll problems have similar chart solutions20 Slab Center-line (x = 0) Temperature Chart Figure 4-15(a)

5 In engel, heat and Mass Transfer21 Chart II Can find T at any x/L from this chart once T at x = 0 is found from previous chart See basis for this chart on the next page = TTTT00 Figure 4-15(b) in engel, heat and Mass Transfer22 Approximate Solutions Valid for for > Values of A1and 1depend on Bi and are different for each geometry (as is Bi) = = 010121rrJeATTTTi = = 01101sin21rrrreATTTTi Cylinder Sphere Slab 11cos21 = = eATTTTi23 Semi-Infinite Solids Plane that extends to infinity in all directions Practical applications: large area for short times Example.

6 Earth surface locallyFigure 4-24 in engel, heat and Mass Transfer24 Multidimensional Solutions Can get multidimensional solutions as product of one dimensional solutions All one-dimensional solutions have initial temperature, Ti, with convection coefficient, h, and environmental temperature, T , starting at t = 0 General rule: twoD= one twowhere oneand twoare solutions from charts for plane, cylinder or sphereFinal ReviewMay 16, 2006ME 375 heat Transfer525 Multidimensional Example()()()slabinfiniteicylinderinfini teicylinderfiniteiTTTtxTTTTtrTTTTtxrT = ,,,,x = a/2x = -a/2 Figure 4-35 in engel, heat and Mass Transfer26 Flow Classifications Forced versus free Internal (as in pipes) versus external (as around aircraft) Entry regions in pipes vs.

7 Fully-developed Unsteady (changing with time) versus unsteady (not changing with time) Laminar versus turbulent Compressible versus incompressible Inviscid flow regions ( not important) One-, two- or three-dimensional27 Flows Laminar flow is layered, turbulent flows are not (but have some structure)Figures 6-9 and 6-16. engel, heat and Mass Transfer28 Boundary Layer Region near wall with sharp gradients Thickness, , usually very thin compared to overall dimension in y directionFigure 6-12 from engel, heat and Mass Transfer29 Thermal Boundary Layer Thin region near solid surface in which most of temperature change occurs Thermal boundary layer thickness may be less than, greater than or equal to that of the momentum boundary layerFigure 6-15.

8 Engel, heat and Mass Transfer30 Dimensionless Convection Nusselt number, Nu = hLc/kfluid Different from Bi = hLc/ksolid Reynolds number, Re = VLc/ = VLc/ Prandtl number Pr = cp/k (in tables) Grashof number, Gr = g TLc3/ 2 g = gravity, = expansion coefficient = (1/ )( / T)p, and T = | Twall T | Peclet, Pe = RePr; Rayleigh, Ra = GrPrFinal ReviewMay 16, 2006ME 375 heat Transfer631 Characteristic Length Can use length as a subscript on dimensionless numbers to show correct length to use in a problem ReD= VD/ , Rex= Vx/ , ReL= VL/ NuD= hD/k, Nux= hx/k, NuL= hL/k GrD= 2 g TD3/ 2, Grx= 2 g Tx3/ 2, GrL= 2 g TL3/ 2 Use not necessary if meaning is clear32 How to Compute h Follow this general pattern Find equations for h for the description of the flow given Correct flow geometry (local or average h?)

9 Free or forced convection Determine if flow is laminar or turbulent Different flows have different measures to determine if the flow is laminar or turbulent based on the Reynolds number, Re, for forced convection and the Grashof number, Gr, for free convection33 How to Compute h Continue to follow this general pattern Select correct equation for Nu (laminar or turbulent; range of Re, Pr, Gr, etc.) Compute appropriate temperature for finding properties Evaluate fluid properties ( , k, , Pr) at the appropriate temperature Compute Nusselt number from equation of the form Nu = C ReaPrbor D Rac Compute h = k Nu / LC34 Property Temperature Find properties at correct temperature Some equations specify particular temperatures to be used ( / w) External flows and natural convection use film temperature (Tw+ T )/2 Internal flows use mean fluid temperature (Tin+ Tout)

10 /235 Key Ideas of External Flows The flow is unconfined Moving objects into still air are modeled as still objects with air flowing over them There is an approach condition of velocity, U , and temperature, T Far from the body the velocity and temperature remain at U and T T is the (constant) fluid temperature used to compute heat transfer36 Flat Plate Flow Equations Laminar flow (Rex, ReL< 500,000, Pr > .6)3/12/12/123/12/12 ==== = Turbulent flow (5x105< Rex, ReL< 107)3 ==== = For turbulent Nu, .6 < Pr < 60 Final ReviewMay 16, 2006ME 375 heat Transfer737 Flat Plate Flow Equations II()3 == = = LLLL wallfkLhNuUC Average properties for com-bined laminar and turbulent regions with transition at xc= 500000 /U Valid for 5x105< ReL< 107and < Pr < 60 Figure 7-10 from engel, heat and Mass Transfer38 heat Transfer Coefficients Cylinder average h (RePr > ; properties at (T + Ts)/25/48/54/13/22/12/1000, + ++==khDNu Sphere average h( Re 80,000; Pr 380; sat Ts; other properties at T )[]4 ++=)