1、Chapter SixIntroduction to Convection6.1 The Convection Transfer ProblemTV,ssTA,qsdAFigure6.1 Local and total convection heat transferThe local heat flux TThqswhere h is the local convection coefficient.Because flow conditions vary from point to point on the surface,both q”and h also vary along the
2、surfacesAssdATTqThe total heat transfer rate by defining an average convection coefficienthTTAhqssThe total heat transfer ratesAsdAqqThe average and local convection coefficients are relatedsAsshdAAh16.2 The Convection Boundary Layers1.The Velocity Boundary LayersuuVelocity boundary layer xFree stre
3、amxyFigure 6.2 Velocity boundary layer on a flat plate.Retardation Shear stresses 1).Boundary Layers thicknessIt is defined as the value of y for which uu99.02).Boundary Layers velocity profileIt refers to the manner in which u varies with y through the boundary layer.uu99.003).Fluid flow composingT
4、he fluid flow is characterized by two distinct regions,a thin fluid layer(the boundary layer)in which velocity gradients and shear stresses are larger and a region outside the boundary layer in which velocity gradients and shear stresses are negligible.4).Friction coefficient22uCsf5).Shear stress0ys
5、yu2.The Thermal Boundary LayerstTTThermal boundary layer xtFree streamxyFigure 6.3 Thermal boundary layer on a flat plate.usT1).Boundary Layers thicknessIt is defined as the value of y for which the ratio .With increasing distance from the leading edge,the effects of heat transfer penetrate further
6、into the free stream and the thermal boundary layer grows.2).The local heat flux The local heat flux,which conduction at y=0,may be obtained by applying Fouriers law to the fluid.That is,3).Heat transfer coefficient Establishing the surface energy balance at the surface,we then obtain the heat trans
7、fer coefficient by combining Equation with Newtons law of cooling.TTTTss0yfyTkqTTyTkhsyf03.Significance of the Boundary layers mctfhyCxlayerboundaryionconcentrathyTxlayerboundarythermalCyuxlayerboundaryvelocitylayersBoundary,Note:For flow over any surface,there will always exit a velocity boundary,a
8、nd hence surface friction.However,a thermal boundary layer exit only if the surface and freetream temperature differ.6.3 Laminar and Turbulent FlowFirst step in treatment of ant Convection problemLaminar boundary layerTurbulent boundary layer?TurbulentregionBuffer layerLaminarSublayerxFigure 6.4 Vel
9、ocity boundary layer development on a flat plate.uuy,vLaminarTransitionTurbulent x,uuvStreamline1.Laminar boundary layerFluid motion is highly ordered and it is possible to identify streamlines along which particles move.Fluid motion along a streamlines is characterized by velocity components in bot
10、h the x and y directions.cx2.Transition boundary layerThe boundary layer is initially laminar,but at some distance from the leading edge,transition to turbulent flow begins to occur.Fluid fluctuations begin to development in the region,and the boundary layer eventually becomes completely turbulent.T
11、he transition to turbulent is accompanied by significant increase in the boundary layer thickness,the shear stress,and the convention coefficients.3.Turbulent boundary layerFluid motion is highly irregular and is characterized by velocity fluctuations.These fluctuations enhance the transfer of momen
12、tum,energy,and hence increase surface friction as well as convention transfer rate.Fluid mixing resulting from the fluctuations make turbulent boundary layer thicknesses larger.Three different regions may be delineated.There is laminar sublayer in which transport is dominated by diffusion and the ve
13、locity profile is nearly linear.There is an adjoining buffer layer in which diffusion and turbulent mixing are comparable,and there a turbulent zone in which transport is dominated by turbulent mixing.Figure 6.5 Variation of velocity layer thickness and transfer coefficient for flow over an isotherm
14、al flat plate.,hLaminar Transition Turbulent sT x xhcx4.Determine the laminar boundary layer distance xcIt is frequently reasonable to assume that transition begins at some location xc.The location is determined by a dimensionless grouping of variables called the Reynolds number,xuxReThe critical Re
15、ynolds number is the value of Rex for transition begins,and for external flow it is known to vary from 105 to 3106,depending on surface roughness.A representative value ofgenerally assumed for boundary calculations.5,105Reccxxu6.4 The Convection Transfer Equations Reason:We can improve our understan
16、ding of the physical effects that determine boundary layers behavior and further illustrate its relevance to convection transport by developing the equations that govern boundary layer conditions.Method:for each of the boundary layers we will identify the relevant physical effects and apply the appr
17、opriate conservation laws to control volumes of infinitesimal size.1.The velocity Boundary LayerMass conservation lawThis law requires that,for steady flow,the net rate at which mass enters the volume(inflow-outflow)must equal zero.udyvdxdxdyyvvdydxxuudydxyx,Figure 6.6 Differential control volume(dx
18、.dy.1)for mass conservation in the two-dimensional velocity boundary layer.0dxdyyvvdydxxuudxvdyu0yvxuThe continuity equationThe continuity equation is a general expression of the overall mass conservation requirement,and it must be satisfied at every point in the velocity boundary layer.Newtons seco
19、nd law of motion:For a differential control volume in the velocity boundary layer,this requirement states that the sum of all forces acting on the control volume must equal the net rate at which momentum leaves and enters the control volume.maF Two kind of forces:Body forces are proportional to the
20、volume.Gravitational,centrifugal,magnetic,and/or electric fields may contribute to the total body force.Surface forces which are proportional to the area are due to the fluid static pressure as well as to viscous stresses.yyyxxxxydxxxxxxdxxxyxydyyyyyydyyyxyxxyFigure 6.6 Normal and shear viscous stre
21、sses for a differential control volume(dx.dy.1)in the two-dimensional velocity boundary layer.dxdyyxpxFyxxxxs,Net surface force for xdxdyxypyFxyxyyys,Net surface force for yFigure 6.7 momentum on the x direction fluxes for a differential control volume(dx.dy.1)in the two-dimensional velocity boundar
22、y layer.dydxuuxuudyuvdxdxdyuvyuvdxuudydxdyux,vy,zNet rate at x-momentumdydxyuvdxdyxuuXyxpxdydxyuvdxdyxuuyxxxx-momentum equationy-momentum equationyvxuxuxx322yvxuyvyy322xvyuyxxyXyxpxyuvxuuyxxxYxypyyvvxvuxyyy Physicals represent Net rate of momentum flow from control volumeNet viscous and pressure for
23、ce,and body forceThe thermal boundary layerdyyadvE,xcondE,xadvE,ycondE,yadvE,dxxadvE,dxxcondE,dyycondE,WgExyzFigure 6.7 Differential control volume(dx.dy.1)for energy conservation in the two-dimensional thermal boundary layer.1).Advected thermal and kinetic energyNeglecting potential energy effects,
24、the energy per unit mass of the fluid includes the thermal internal energy e and the kinetic energy V2/2.dxdyVeuxdydxVeuxVeudyVeuEEdxxadvxadv22222222,2).ConductionEnergy is also transferred across the control surface by molecular processes.dxdyxTkxdydxxTkxxTkdyxTkEEdxxcondxcond,3).Work interactionsE
25、nergy may also be transferred to and from the fluid in the control volume by work interactions involving the body and surface forces.dxdyuydxdyupxdxdyXuWyxxxxnet,Thermal Energy Equation02222qvuyvuxpvypuxYvXuyTkyxTkxVevyVeuxyyyxxyxxpeiqypvxpuyTkyxTkxyivxiuNet advected energy at x and yNet conduction
26、energy at x and yNet work energy at x and y2222322yvxuyvxuxvyuExample there are few situations for which exact solutions to the convection transfer equations may be obtained.What is termed parallel flow.In this case gross fluid motion is only in one direction.Consider a special case of parallel flow
27、 involving stationary and moving plates of infinite extent separated by a distance L,with the intervening space filled by an incompressible fluid.1.What is the appropriate from of the continuity equation?2.Beginning with the momentum equation,determine the velocity distribution between the plates.3.
28、Beginning with the energy equation,determine the temperature distribution between the plates.4.Consider conditions for which the fluid is engine oil with L=3mm.The speed of the moving plate is U=10m/s,and the temperatures of the stationary and moving plates are T0=10 and TL=30,respectively.Calculate
29、 the heat flux to each of the plates and determine the maximum temperature in the oil.Moving plateParallel flowEngine oilvy,ux,0vCTL030CT0010Stationary platemmL3smU/10Schematic:Assumptions:1.Steady-state conditions.2.Two-dimensional flow(no z).3.Incompressible fluid with constant properties.4.No bod
30、y forces.5.No internal energy generation.Properties:engine oil:888.2kg/m3,k=0.145W/m.K,10-6m2/s,=0.799N.s/m2.Moving plateParallel flowEngine oilvy,ux,0vCTL030CT0010Stationary platemmL3smU/10Schematic:Assumptions:1.Steady-state conditions.2.Two-dimensional flow(no z).3.Incompressible fluid with const
31、ant properties.4.No body forces.5.No internal energy generation.Properties:engine oil:888.2kg/m3,k=0.145W/m.K,10-6m2/s,=0.799N.s/m2.Analysis:1.For an incompressible fluid and parallel flow,it means constant number0v0yvxu0 xuu velocity is independent of x.Velocity field is full developed.2.For two-di
32、mensional,steady-state conditions with v=0,X=0 0 xuXyxpxyuvxuuyxxxypxyxxx0Motion of fluid is sustained not by the pressure gradient,but by an external force that provides for motion of the top plate relative to the bottom plate.0 xp0322yvxuxuxxyyx0 xuyx022xuMoving plateParallel flowEngine oilvy,ux,0
33、vCTL030CT0010Stationary platemmL3smU/10Schematic:Assumptions:1.Steady-state conditions.2.Two-dimensional flow(no z).3.Incompressible fluid with constant properties.4.No body forces.5.No internal energy generation.Properties:engine oil:888.2kg/m3,k=0.145W/m.K,10-6m2/s,=0.799N.s/m2.Analysis:1.For an i
34、ncompressible fluid and parallel flow,it means constant number0v0yvxu0 xuu velocity is independent of x.Velocity field is full developed.2.For two-dimensional,steady-state conditions with v=0,X=0 0 xuXyxpxyuvxuuyxxxypxyxxx0Motion of fluid is sustained not by the pressure gradient,but by an external
35、force that provides for motion of the top plate relative to the bottom plate.0 xp0322yvxuxuxxyyx0 xuyx022xu3.The energy equation may be simplified for the prescribed conditions.0,0,0,0qxpxuvqypvxpuyTkyxTkxyivxiu2yuuyTkyxTkxxiuBecause the top and bottom plates are at uniform temperatures,the temperat
36、ure field must be fully developed,in which case .The enthalpy is function of temperature and pressure,it be expressed as0 xT0 xppixTTixi2220yuuyTk222yuuyTk6.5 Approximation and Special Conditions Simplify the forms of equationsIt is a rare situation when all of the terms need to be considered,and it
37、 is customary to work eith simplified forms of the equation.Usual situation(two-dimension)Steady(time-independent)Incompressible(constant)Constant properties(k,etc)Negligible body forces(X=Y=0)Without energy generation(q=0)Boundary approximationsBoundary layer approximationsxvyvxuyuvu,xTyTVelocity b
38、oundary layerThermal boundary layer0322yvxuxuxx0322yvxuyvyyyuxvyuyxxyBasing on foregoing simplifications and approximationsMathematical model for the convection transfer in different boundary layersEquations may be solved to determine the spatial variations of u,v,T in the different boundary layers.
39、For incompressible,constant property flow,equations(1)and(2)are uncoupled from(4).That is,it may be solved for the velocity field.u(x,y)and v(x,y).Then the velocity gradient could be evaluated,and the wall shear stress could be obtained.Through the appearance of u and v in equation(4),the temperatur
40、e is coupled to the velocity field.The convection heat coefficient may be determined.0yvxu221yuvxpyuvxuu0yp222yucyTyTvxTup 1 2 3 4Note:02yucpIn most situation viscous may be neglected relative to other terms.In fact it is only for sonic flow or the high speed motion of lubricating oils.0ypThe pressu
41、re does not vary in the direction normal to the surface.It in the boundary layer depends only on x and is equal to the pressure in the freestream outside the boundary layer.It be treated as known quantity.Purpose:1.One major motivation has been to cultivate an appreciation for the different physical
42、 processes.These processes will affect wall friction,as well as energy transfer in boundary layers.2.A second motivation arises from the fact that the equations may be used to identify key boundary layer similarity parameters,as well as important analogies between momentum and heat transfer.6.6 Boun
43、dary Layer Similarity:The Normalized Convection Transfer Equations 221yuvxpyuvxuu22yuvyuvxuu222yucyTyTvxTup22yTyTvxTuStrong similarity Same formThis equation describes low-speed,forced convection flows,which are found in many engineering applications.22yyvxu,:equationuu,:equationTTAdvection termsDif
44、fusion termNondimensionalizing Implications of this similarity may be developed in a rational manner by first nondimensionalizing the governing equations.Boundary layer similarity parametersIndependent dimensionless variablesLyyLxxandCharacteristic lengthDependent dimensionless variablesVvvVuuandVelocity upstream of the surfacessTTTTT2Vpp22yuVLvxpyuvxuu22yTVLyTvxTuVLReLVLVLPrReLPrSimilarity parametersReynolds Number:VLReLPrandtl Number:PrDimensionless boundary layer equations:221yuRexpyuvxuuL221yTPrReyTvxTuL0yvxu
