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1、Chapter SixIntroduction to Convection6.1 The Convection Transfer ProblemFigure6.1 Local and total convection heat transferThe local heat flux where 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 surfaceThe tota
2、l heat transfer rate by defining an average convection coefficientThe total heat transfer rateThe average and local convection coefficients are related6.2 The Convection Boundary Layers1.The Velocity Boundary LayersVelocity boundary layerFree streamxyFigure 6.2 Velocity boundary layer on a flat plat
3、e.Retardation Shear stresses 1). Boundary Layers thicknessIt is defined as the value of y for which 2). Boundary Layers velocity profileIt refers to the manner in which u varies with y through the boundary layer. 3). Fluid flow composingThe fluid flow is characterized by two distinct regions, a thin
4、 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 coefficient5). Shear stress2. The Thermal Boundary LayersThermal boundary layerFree streamxyF
5、igure 6.3 Thermal boundary layer on a flat plate.1). 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 into the free stream and the thermal boundary layer grows. 2). The local
6、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 transfer coefficient by combining Equation with Newtons law of cooling.
7、 3. Significance of the Boundary layersNote: For flow over any surface, there will always exit a velocity boundary, and 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 Con
8、vection problemLaminar boundary layerTurbulent boundary layer?TurbulentregionBuffer layerLaminarSublayerxFigure 6.4 Velocity boundary layer development on a flat plate.y, vLaminarTransitionTurbulent x, uuvStreamline1. Laminar boundary layerFluid motion is highly ordered and it is possible to identif
9、y streamlines along which particles move. Fluid motion along a streamlines is characterized by velocity components in both the x and y directions.2. Transition boundary layerThe boundary layer is initially laminar, but at some distance from the leading edge, transition to turbulent flow begins to oc
10、cur. Fluid fluctuations begin to development in the region, and the boundary layer eventually becomes completely turbulent . The transition to turbulent is accompanied by significant increase in the boundary layer thickness, the shear stress, and the convention coefficients.3. Turbulent boundary lay
11、erFluid motion is highly irregular and is characterized by velocity fluctuations. These fluctuations enhance the transfer of momentum, energy, and hence increase surface friction as well as convention transfer rate. Fluid mixing resulting from the fluctuations make turbulent boundary layer thickness
12、es larger. Three different regions may be delineated. There is laminar sublayer in which transport is dominated by diffusion and the velocity 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 trans
13、port is dominated by turbulent mixing.Figure 6.5 Variation of velocity layer thickness and transfer coefficient for flow over an isothermal flat plate.Laminar Transition Turbulent 4. Determine the laminar boundary layer distance xcIt is frequently reasonable to assume that transition begins at some
14、location xc. The location is determined by a dimensionless grouping of variables called the Reynolds number,The critical Reynolds 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 ofgen
15、erally assumed for boundary calculations.6.4 The Convection Transfer Equations Reason: We can improve our understanding 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
16、conditions.Method:for each of the boundary layers we will identify the relevant physical effects and apply the appropriate 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 mas
17、s enters the volume (inflow - outflow) must equal zero.Figure 6.6 Differential control volume (dx . dy . 1) for mass conservation in the two-dimensional velocity boundary layer.The continuity equationThe continuity equation is a general expression of the overall mass conservation requirement, and it
18、 must be satisfied at every point in the velocity boundary layer.Newtons second 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
19、the control volume. Two kind of forces:Body forces are proportional to the 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 stress
20、es.xyFigure 6.6 Normal and shear viscous stresses for a differential control volume (dx.dy.1) in the two-dimensional velocity boundary layer.Net surface force for xNet surface force for yFigure 6.7 momentum on the x direction fluxes for a differential control volume (dx.dy.1)in the two-dimensional v
21、elocity boundary layer.Net rate at x-momentumx-momentum equationy-momentum equation Physicals represent Net rate of momentum flow from control volumeNet viscous and pressure force, and body forceThe thermal boundary layerxyzFigure 6.7 Differential control volume (dx . dy . 1) for energy conservation
22、 in the two-dimensional thermal boundary layer.1). Advected thermal and kinetic energyNeglecting potential energy effects, the energy per unit mass of the fluid includes the thermal internal energy e and the kinetic energy V2/2.2). ConductionEnergy is also transferred across the control surface by m
23、olecular processes.3). Work interactionsEnergy may also be transferred to and from the fluid in the control volume by work interactions involving the body and surface forces.Thermal Energy EquationNet advected energy at x and yNet conduction energy at x and yNet work energy at x and yExample there a
24、re 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 involving stationary and moving plates of infinite extent separated by a
25、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. Beginning with the energy equation, determine the temperature distrib
26、ution 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 the heat flux to each of the plates and determine the maximum
27、temperature in the oil.Moving plateParallel flowEngine oilStationary plateSchematic: 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/
28、m.K,10-6m2/s, =0.799N.s/m2.Moving plateParallel flowEngine oilStationary plateSchematic: 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=0.1
29、45W/m.K,10-6m2/s, =0.799N.s/m2.Analysis:1. For an incompressible fluid and parallel flow, it means constant numberu velocity is independent of x. Velocity field is full developed. 2. For two-dimensional, steady-state conditions with v=0, , X=0 Motion of fluid is sustained not by the pressure gradien
30、t, but by an external force that provides for motion of the top plate relative to the bottom plate.Moving plateParallel flowEngine oilStationary plateSchematic: Assumptions:1. Steady-state conditions.2. Two-dimensional flow (no z).3. Incompressible fluid with constant properties.4. No body forces.5.
31、 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 numberu velocity is independent of x. Velocity field is full developed. 2. For two-dimensional, steady-state conditions
32、with v=0, , X=0 Motion 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.3. The energy equation may be simplified for the prescribed conditions.Because the top and bottom plates are at uniform temperatu
33、res, the temperature field must be fully developed, in which case . The enthalpy is function of temperature and pressure, it be expressed as6.5 Approximation and Special Conditions Simplify the forms of equationsIt is a rare situation when all of the terms need to be considered, and it is customary
34、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 approximationsVelocity boundary layerThe
35、rmal boundary layerBasing 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. For incompressible, constant property flow, equ
36、ations (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 temperature is coupled to the velocity
37、field. The convection heat coefficient may be determined. Note: In 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.The pressure does not vary in the direction normal to the surface. It in the boundary lay
38、er 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 processes. These processes will affect wall friction, as well as energy tr
39、ansfer 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 Boundary Layer Similarity: The Normalized Convection Transfer Equations St
40、rong similarity Same formThis equation describes low-speed, forced convection flows, which are found in many engineering applications.Advection termsDiffusion termNondimensionalizing Implications of this similarity may be developed in a rational manner by first nondimensionalizing the governing equations.Boundary layer similarity parametersIndependent dimensionless variablesCharacteristic lengthDependent dimensionless variablesVelocity upstream of the surfaceSimilarity parametersReynolds Number:Prandtl Number:Dimensionless boundary layer equations:
