Presentation on theme: "CHAPTER 5 Principles of Convection"— Presentation transcript:
1 CHAPTER 5 Principles of Convection 5-1 INTRODUCTION5-2 VISCOUS FLOWShear stress:boundary layer : the region of flow that develops from the leading edge of a plate in which the effects of the viscosity are observedThe outside boundary of a boundary layer is usually chosen as the point where the velocity of flow is 99 percent of the free-stream value.
2 Three regimes of boundary-layer flow 1. Laminar flow2. Transitional flow3. Turbulent flowThe transition occurs whenRenolds numberFor most analytical purposes, the critical number for the transition is usually taken as
3 The critical Re for transition is strongly dependent on the surface roughness condition and the “turbulent level”of the free-stream.The normal range for the beginning of transition is between :For very large disturbances present in the flow, transition may begin with Renolds number as low asFoe flows that are very free from fluctuation, the transition may not start untilThe transition is completed at Re twice the value at the transition begin.
4 The relative shape for the velocity profiles in laminar and turbulent flow The laminar profile is approximately parabolicStructure of turbulent profile :Laminar sublayer that is nearly linear.Turbulent portion which is relatively flat in comparison with the laminar profile.
5 The physical mechanism of viscosity in fluids In laminar flow, the viscosity is attributed to the exchange of momentum between different laminas by the movement of molecules.For a gasIn turbulent flow, the momentum exchange between different layers is caused by the macroscopic movement of fluid chunks. We can expect a larger viscous-shear in turbulent flow than in laminar flow, due to which the velocity profile is flat in a turbulent boundary layer.
6 Flow in a tubeThe critical ReThe range of Re for transition isContinuity relation in a tube isRe based on mass velocity is defined as
7 5-3 INVISCOUS FLOWThe Bernoulli equation for flow along a stream results:In differential form,The energy equation for compressible fluidi is the enthalpy defined by
8 Equation of state of fluid Relations applicable to reversible adiabatic flow:
9 5-4 LAMINAR BOUNDARY LAYER ON A FLAT PLATE Assumptions:Incompressible and steady flowNo pressure variation in the direction perpendicular to the plate.Constant viscosityViscous-shear in y direction is negligible.Two methods to study motion of fluid.1. Newton’s law of motion2. Force balanceWhich applies to a elemental control volume fixed in space.Which applies to a system of constant mass.
10 Mass continuity equation Mass in the left face isMass out of the left face isMass in the bottom face isMass out of the top face isMass balance on the element isMass continuity equation
11 Derivation of momentum equation Mass in the left face isMomentum flux in the left face isMomentum flux out of the left face isMass in the bottom face isMomentum flux in the x direction entering the bottom faceMass out of the top face isMomentum flux in the x direction leaving the top face
12 Pressure forces on the left and right faces are and Net pressure force in the direction of motion isViscous-shear force on the bottom face isViscous-shear force on the top face isBalancing force and momentum in x direction givesFinal result——Momentum equation
13 Integral momentum equation of the boundary layer. Mass flow through plane 1[a]Momentum flow through plane 1[b]Momentum flow through plane 2[c]Mass flow through plane 2[d]Carried momentum in x direction by the flow through plane A-AThe net momentum flow out of the control volume is
14 By the use oforThe pressure force on plane 1 isThe pressure force on plane 2 isThe shear force at wall isSetting the force on the element equal to the net increase in momentum gives——Integral momentum equation of the boundary layer.
15 For constant pressure，from Bernoulli equation One obtainsIntegral momentum equation of the boundary layer becomes
16 Evaluation of boundary layer thickness Boundary conditions areInserting the expression into equation [5-17]For constant-pressure conditionCarrying out the integration leads tosupposeAppling boundary conditions obtainsSeparation of variables leads to
17 , so thatIn terms of Renolds numberExact solution
18 5-5 ENERGY EQUATION OF THE BOUNDARY LAYER Assumptions:incompressible steady flowConstant viscosity ,thermal conductivity, and specific heatNegligible heat conduction in the direction of flowEnergy convected in left face+ energy convected in bottom face+ energy conducted in bottom face+net viscous work done on element= energy convected out right face+ energy out top face+heat conducted out top face
19 ［5-12］ ［5-22］ The viscous shear force over dx The distance through which the force moves in respect to the control volume dxdy isThe net viscous energy delivered to the element is［5-12］Energy balance corresponding to the quantities shown in figure 5-6 is［5-22］UsingAnd dividing by gives——Energy equation of the laminar boundary layer.
20 Order-of-magnitude analysis ［5-25］［5-23］［5-26］A striking similarity between [5-25] and [5-26]［5-24］
21 5-6 THE THERMAL BOUNDARY LAYER 2. Definition of h［5-27］［5-28］［5-29］3. Temperature distribution in the thermal boundary layerBoundary conditionsAt y=0［a］At［c］At［b］At y=0［d］
22 Conditions (a) to (d) may be fitted to a cubic polynomial ［5-30］4. Integral energy equation of the boundary layerEnergy convected in +viscous work within element +heat transfer at wall=energy convected outThe energy convected through plane 1 isThe energy convected out through plane 2 isThe net viscous work done within element isThe mass flow through plane A-AHeat transfer at wallThe energy carried with is
23 ［5-32］ Combining the above energy quantities gives ——integral energy equation of the boundary layer.5. Thermal boundary layer thicknessInserting (5-30) and (5-19) into (5-32) gives［5-30］
24 Assume thermal boundary layer is thinner than the hydrodynamic boundary layer Making substitution5-335-34Neglecting givesPerforming the differentiation givesorBut according to page 217and
25 so that we have5-35Noting thatSolution isWhen the boundary conditionatatis applied , the final solution become5-36where5-37When the plate is heated over the entire length5-38
26 6. Prandtl number(see page 225) 5-397. Nusselt number5-40Substituting (5-21) and (5-36) gives5-41Nusselt number5-42Finally,5-43For the plate heated over its entire length5-44
27 9. Average heat transfer coefficient and Nusselt number 5-45or5-46a5-46bwhereFilm temperature5-47
28 10. Constant heat flux 11. Other relations or 5-485-495-50or11. Other relationsFor laminar flow on an isothermal flat plateFor Rex Pr＞1005-51For the constant-heat-flux case, is changed to , and is replaced by
29 Basic laws for inviscous flow Velocity and temperature distributionsMass continuity equationResults:Momentum equationIntegral momentum equationEnergy equationA striking similarityIntegral energy equation
30 5-7 THE RELATION BETWEEN FLUID FRICTION AND HEAT TRANSFER [5-52]The shear stress isUsing the velocity distribution given by equation(5-19), we haveMaking use of the relation for the boundary-layer thickness givesCombining (5-52) and (5-53) leads to
31 The exact solution is[5-44] may be rewritten asBy introduction of Stanton number——Reynolds-Colburn analogy
32 5-8 TURBULENT-BOUNDARY-LAYER HEAT TRANSFER Structure of turbulent flow:Laminar sublayerBuffer layerTurbulentThe physical mechanism of heat transfer in turbulent flow is similar to that in laminar flow.Difficulty: there is no completely adequate theory to predict turbulent-flow behaviorvelocity fluctuation in a turbulent flow
33 Shear stress giving rise to velocity fluctuations in turbulent flow Eddy Viscosity and Mixing Length ( 湍流粘度与混合长度 )Mean free path and Prandtl mixing lengthPrandtl postulated:
35 Universal velocity profile For fully turbulent regionFrom equation [5-65], we haveSubstituting this relation along with equation (5-64) into equation (5-63) givesorSubstituting this relation into Eq (5-69) for and integrating givesUniversal velocity profileLaminar sublayer : 0<y+<5Buffer layer : 5<y+<30Turbulent layer : 30<y+<400For fully turbulent regionFor regions where both molecular and turbulent energy transport are important
36 Turbulent Heat Transfer Based on Fluid-Friction Analogy 3. Average-friction coefficient for a flat plate:A simpler formula for lower Reynolds number is2. The local skin-friction coefficient over a flat plate:Table 5-14. Local turbulent heat transfer coefficient
37 5. Average heat transfer coefficient over the entire laminar-turbulent boundary layer For higher Reynolds number,using equation (5-79), one obtainsFrom , the above equation can be rewritten as6. Equation suggested by WhitakerAlternatively,for the laminar portionfor the turbulent portionConstant Heat FluxOne obtains
38 TURBULENT BOUNDARY LAYER THICKNESS 1. Velocity profile in a turbulent boundary layerThe first case: The boundary layer is fully turbulent from the leading edge of the plate:2. Shear stress at wallThe second case: The boundary layer follows a laminar growth pattern up toand a turbulent growth thereafterSo that3. Integrating the integral momentum equationIntegrating [5-90] leads toCombining the above various relations givesIntegrating and clearing terms gives
39 (a) Semilog scaleBoundary-layer thickness for atmospheric air at u=30m/s.(b) log scale
40 5-10 HEAT TRANSFER IN LAMINAR TUBE FLOW 1. Velocity distribution5-98
41 2. Energy balance analysis and temperature distribution Net energy convected out = net heat conducted inwhich may be rewritten
42 5-98assumeB.C:Inserting Eq (5-98) into Eq (5-99)
43 Bulk temperature 4. Convection heat transfer coefficient 1. Definition of convection heat transfer coefficient in tube flowLocal heat flux =2. Bulk temperature ( 整体温度 )3. Wall temperature
44 5-11 TURBULENT FLOW IN A TUBE For laminar flowassumeIntegrating (a)For turbulent flowHeat transfer at wall is
45 Substituting ( B ) and ( C ) into ( A ) gives Reynolds analogy for tube flowHeat transfer at wall is( B )Shear stress at wall isThe pressure drop can be expressed in terms of friction factor( D ) is modified by PrSo thatA more correct relation( C )