课件参考传热学chap

1、5-1 IntroductionConvection: heat transfer between a surface and a flowing fluid. Newtons law of cooling: q = hATWhere h is the convection heat transfer coefficient.Target of study: the way of predicting the value of h.There are different kinds of convection:Chapter 5 Principles of ConvectionConvecti

2、onWith phase change有相变Without phase change无相变Forced convection强制对流(Flow due to external force)Free convection自然对流(Flow due to density differenceMixed convection混合对流Boiling 沸腾Condensation 凝结Relation between h and the temperature fieldThere is no-slip (无滑移) requirement for viscous fluid (粘性流体) flowing

3、 over a wall, that is the velocity at the wall is zero, the heat transfer mode is then conduction.The Fouriers Law is applicable:xyuwhere k is the heat conductivity of the fluid; is the temperature gradient in the fluid at the position of the wall; asThis is an important relation between h and the t

4、emperature field in the fluid.5-2 Viscous (粘性) FlowThe region of flow which develops from the leading edge of the plate in which the effects of viscosity are observed is called the boundary layer (边界层) or hydrodynamic boundary layer (运动边界层). Using the dynamic viscosity , the viscous force inside the

5、 boundary layer can be described as:xyuuLaminar regionTransitionTurbulentxcLaminar sublayerUsually boundary layer is defined to end at a y position where the velocity es 99% of the free stream value.Initially, the boundary layer development is laminar (层流);(层流底层)Initially, the boundary layer develop

6、ment is laminar (层流); At some critical distance (or Reynolds number Rex):xyuuLaminar regionTransitionTurbulentxcLaminar sublayeru = free-stream velocityx = distance from leading edge = / = kinematic viscousity(层流底层)a transition (过渡) process takes place until the flow es turbulent (湍流). For turbulent

7、 flow there is also a laminar sublayer.For flow inside a tube, a boundary layer develops at the entrance. Eventually, the boundary layer fills the entire tube, and the flow is said to be fully developed.If the flow is laminar, a parabolic velocity profile is experienced.The Reynolds number is again

8、used as criterion for laminar and turbulent flow, if:Flow is usually no more laminar.If the flow is turbulent, a somewhat blunter (较钝的) profile is observed.A range of Reynolds numbers for transition may be observed depending on the pipe roughness and smoothness of the flow:Laminar sublayerTurbulent

9、core5-3 Inviscid (无粘) FlowAlthough no real fluid is inviscid, in some cases the flow may be treated as such. For example, in the flat plate problem discussed above, the flow above the boundary layer will behave as a nonviscous flow system. This is because the shear stress isxyuuLaminar regionTransit

10、ionTurbulentxcLaminar sublayerand above the boundary layer, there is no velocity gradient, which yields = 0For inviscid flow, there is the Bernoulli equation for flow along a streamline:5-4 Laminar Boundary Layer on a Flat PlateFor 2-D pressible flow with constant properties, the general form of con

11、servation equations are:1. Continuity equation:In x-direction:In y-direction:Mass change:Now the mass balance is:Continuity equation:For 2-D steady pressible flow with constant properties:2. Momentum equation:Force = mass acceleration（F=d(mV)/d）Momentum in in x-direction:Momentum out in x-direction:

12、Momentum in in y-direction:Momentum out in y-direction:The final Momentum equation is:3. Energy equation:If u=v=0, it reduces to heat conduction equation:However, analytical solution can only be obtained for a few simple problems. 2. Momentum equation:3. Energy equation:1. Continuity equation:From f

13、luid dynamics, we know for laminar boundary layer of steady flow, the momentum equation can be simplified:Continuity equation:Momentum equation:Analytical solutions to the laminar flow above a flat plate was obtained first by Blasius (Appendix B, p655, 8th edition), from which the thickness of the b

14、oundary layer is:However, analytical solution can only be obtained for a few simple problems. In some cases, the approximate solution of integration method (积分法) has advantages.Integration method:(1) The conservation equations are satisfied for the whole control volume (integration), instead of each

15、 element, in the boundary layer;(2) Usually the velocity and temperature distributions are assumed to be polynomials in the boundary layer.From Fluid dynamics, the integration form of the momentum equation is:For laminar flow over a flat plate, we have the following boundary conditions:xyuAnother B.

16、C. can be obtained from the momentum equation for a constant pressure condition:We choose a polynomial function for the velocity with four constants:u = C1 + C2 y + C3 y2 + C4 y3 The 4 constants can be obtained from the 4 B.Cs., which yields:From the two equations one has:xyuWhich is very close to t

17、he exact solution:The solution is:5-5 Energy Equation of the Boundary LayerThe energy equation is developed here. Three assumptions:1) pressible steady flow;2) Constant fluid property;3) Negligible conduction in x direction.The energy balance for the element is:Energy convected in left face +Energy

18、convected in bottom face +Heat conducted in bottom face +Net viscous work done on elementxyxyu=Energy convected out right face +Energy convected out top face +Heat conducted out top faceEnergy convected in left face:Energy convected out right face:Energy convected in bottom face:xyuvHeat conducted i

19、n bottom face:Energy convected out top face:Heat conducted out top face:The net viscous work done on element can be computed as a product of the viscous force and the distance it moves.xyuvThe net viscous work done on element can be computed as a product of the viscous-shear force and the distance i

20、t moves.The viscous-shear force is the product of the stress and the area dx:The distance should be the difference of the velocity of the up and bottom face by unit time:So the net viscous work done on the element is:According to the quantities, after neglecting the second order differentials, the e

21、nergy balance will be:The energy balance will be:asThe final energy balance is:The magnitude of the second term on the right side is small compared with other terms when low-velocity flow is studied, and is thus neglected. This can be shown with an order of magnitude analysis:The second term on can

22、be neglected if the ratio of the two terms is small, that is:Define the Prandtl number as:An example of the ratio is 0.012 for air at u=70m/s with:p = 1 atm, T = 20C, Pr = 0.7, cp = 1005 J/kgSo the energy equation of the boundary layer is:Then the requirement is:Compare:The solution of the two equat

23、ions will have the same form if Pr = 1.5-6 The Thermal Boundary Layer1. Thermal boundary layerHydrodynamic boundary layer: the region of the flow where viscous force are felt;Thermal boundary layer: the region where temperature gradients are presented. The thermal boundary layer is defined to end at

24、 a y position where the temperature es 99% of the free stream value. There is no slip requirement for viscous fluid flowing over a wall, that is the velocity at the wall is zero, the heat transfer mode is then conduction.So we need to know the temperature field.xyTutuT2. The integration form of ener

25、gy equationNow we use the method of integration to find the integration form of energy equation.From the energy equation:Given an x, from y = 0 to y = perform integration to the above equation: xyTutuTUse the method of partial integration(分部积分):in which v can be represented by u from the equation:Th

26、e diffusion term is: Then: The left side of the above equation can be simplified further as:The final integral energy equation is:The integral momentum equation is:Equivalent to the one in the textbook:For thermal boundary layer, we have the following boundary conditions:Another B.C. can be obtained

27、 from the energy equation:We choose a polynomial function for the temperature with four constants:T = C1 + C2 y + C3 y2 + C4 y3 The 4 constants can be obtained from the 4 B.Cs., which yields:xyTutuT3. The Convection heat Transfer CoefficientConsider a plate with an unheated length of x0. The hydrody

28、namic boundary layer develops from the leading edge. From the Energy Equation:Let = t /xyTutx0Assume t Then 4 t For liquid metals, as is very big, Pr 0.01, the thermal boundary layer will go very far; tFor air: Pr 1, t For the result we have obtained:xyTutx0An assumption has been made that t ). The

29、assumption is satisfactory for fluids having Pr 0.7.Most gases and liquids fall in this category.For liquid metal, as Pr has an order of 0.01, the result is not valid.Finally:Introduce Nusselt number (Wilhelm Nusselt):xyTutx0If x0 = 0, then:Which is identical as exact solution.Average heat transfer

30、coefficient and Nusselt number:xyTutTwThe result is based on the assumption that the fluid properties are constant. However, they are not. It is mended that the properties are calculated at an average temperature Tf, called film temperature:Here Tw is constant as we are discussing constant wall temp

31、erature, or isothermal surface.5. Constant Heat FluxAfter discussing constant wall temperature or isothermal surface, lets consider a plate with constant heat flux. The objective is to find the distribution of the plate surface temperature.It can be shown that for constant heat flux:xyTutqw = constw

32、here:The average temperature difference will be:6. Other RelationsThe results we have obtained so far are for fluids with Prandtl number between about 0.6 to 50. They can not be applied to liquid metals (Pr 0.01) and heavy oils or silicones (Pr 50). For a wide range of Prandtl numbers, Churchill fou

33、nd the following relation for laminar flow over an isothermal flat plate:For constant heat flux case:0.3387 0.46370.0468 0.02075-97 Air at 1 atm and 300 K blows across a 10 cm square plate at 30 m/s. Heating does not begin until x = 5.0 cm, after which the plate surface is maintained at 400 K. Calcu

34、late the total heat lost by the plate.Solution:xyTutx0From Table A-5 (p514): Pr = 0.697Flow is laminar.xyTutx0The integration can be calculated approximately:0.050.1xx0.0550.0650.0750.0850.095f(x)Dx11.041287.3964276.0586575.2801944.748386If x = 0.005x0.05250.0575 0.0625 0.0675 0.0725 0.0775 0.0825 0

35、.0875 0.0925 0.0975 f(x)Dx7.0230264.7785723.9617233.4859223.1593442.9151082.7225042.5650282.4328532.319677If x = 0.0025, q = 35.9 Wx0.051250.05375 0.05625 0.05875 0.06125 0.06375 0.06625 0.06875 0.07125 0.07375 f(x)Dx4.4464483.0526412.550562.259562.0602121.9111261.7934051.6969521.6157791.546064x0.07

36、625 0.07875 0.08125 0.08375 0.08625 0.08875 0.09125 0.09375 0.09625 0.09875 f(x)Dx1.485231.4314611.3834331.3401531.3008591.2649521.2319561.2014861.1732241.14691If x = 0.00125, q = 36.2 Wf(x)5-98 Air at 1 atm and 300 K blows across a 10 cm square plate at 30 m/s. A 0.5cm strip centered at x = 6.0 cm

37、is heated at 400 K. Calculate the total heat lost by the strip.Solution:xyTutx0From Table A-5 (p514): Pr = 0.697Flow is laminar.xyTutx0q=296.30.0050.1(400-300) =14.8 W 0.059750.065-7 The Relation between Fluid Friction and Heat TransferWe have seen that the temperature and flow field are related. De

38、fine the friction coefficient Cf :Also:from:where:so that:Compare the above two equations and neglect the difference between 0.323 and 0.332 we have:This equation is called the Reynolds-Colburn analogy, which expresses the relation between fluid friction and heat transfer for laminar flow on a flat

39、plate.Thus the heat transfer coefficient could be determined by making measurements of the friction drag on a plate under isothermal conditions.It turns out that the equation can be applied to turbulent flow over a flat plate and in a modified way to turbulent flow in a tube;It does not apply to lam

40、inar tube flow.5-8 Turbulent Boundary Layer Heat TransferConsider a portion of a turbulent boundary layer. A very thin region near the plate surface has a laminar character, and the viscous action and heat transfer take place under circumstances like those in laminar flow.Further out, at larger y di

41、stances from the plate, some turbulent action is experienced, but the molecular viscous action and heat conduction are still important.Still further out, the flow is fully turbulent.xyuuLaminar sublayerBuffer layerTurbulent1. Eddy diffusivity (紊流扩散率)If one observes the instantaneous macroscopic velo

42、city in a turbulent flow, significant fluctuations about the mean flow velocity are observed.Time averaged equations:For 2-D pressible turbulent flow, the momentum and energy equations are:Compared to the equations in laminar flow, here we have additional terms related to the product of fluctuation

43、quantities.Those terms are called normal stress and shear stress due to fluctuationEddy diffusivity (紊流扩散率)In turbulent flow, one may assume the total heat conduction as a sum of the molecular conduction and macroscopic eddy conduction:where h is the eddy diffusivity of heat (紊流热扩散率).In a similar fashion, the shear stress can be written as:where m is the eddy diffusivity of momentum (紊流动量扩散率).2. Heat transfer based on fluid-friction analogyFor laminar flow over a flat plate, there is the Reynolds-Colburn analogy (比拟):If we define Prt = m/m.If we expect that the eddy momentum and energy tran