John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 57
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Klein. Heat transfer to laminarflow in a round tube or a flat plate—the Graetz problem extended.Trans. ASME, 78:441–448, 1956.[7.8] M. S. Bhatti and R. K. Shah. Turbulent and transition flow convective heat transfer in ducts. In S. Kakaç, R. K. Shah, and W. Aung,editors, Handbook of Single-Phase Convective Heat Transfer, chapter 4.
Wiley-Interscience, New York, 1987.[7.9] F. Kreith. Principles of Heat Transfer. Intext Press, Inc., New York,3rd edition, 1973.[7.10] A. P. Colburn. A method of correlating forced convection heattransfer data and a comparison with fluid friction. Trans. AIChE,29:174, 1933.[7.11] L. M. K. Boelter, V.
H. Cherry, H. A. Johnson, and R. C. Martinelli.Heat Transfer Notes. McGraw-Hill Book Company, New York, 1965.[7.12] E. N. Sieder and G. E. Tate. Heat transfer and pressure drop ofliquids in tubes. Ind. Eng. Chem., 28:1429, 1936.[7.13] B. S. Petukhov. Heat transfer and friction in turbulent pipe flowwith variable physical properties. In T.F. Irvine, Jr. and J. P. Hartnett, editors, Advances in Heat Transfer, volume 6, pages 504–564.Academic Press, Inc., New York, 1970.[7.14] V. Gnielinski. New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chemical Engineering, 16:359–368,1976.References[7.15] D.
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Shah, and W. Aung, editors, Handbook of Single-Phase Convective Heat Transfer, chapter 17. Wiley-Interscience, New York,1987.[7.20] B. Lubarsky and S. J. Kaufman. Review of experimental investigations of liquid-metal heat transfer. NACA Tech. Note 3336, 1955.[7.21] C. B. Reed. Convective heat transfer in liquid metals. In S. Kakaç,R. K. Shah, and W. Aung, editors, Handbook of Single-Phase Convective Heat Transfer, chapter 8. Wiley-Interscience, New York, 1987.[7.22] R. A. Seban and T. T. Shimazaki. Heat transfer to a fluid flowingturbulently in a smooth pipe with walls at a constant temperature.Trans. ASME, 73:803, 1951.[7.23] R.
N. Lyon, editor. Liquid Metals Handbook. A.E.C. and Dept. of theNavy, Washington, D.C., 3rd edition, 1952.[7.24] J. H. Lienhard. Synopsis of lift, drag, and vortex frequencydata for rigid circular cylinders. Bull. 300. Wash. State Univ.,Pullman, 1966. May be downloaded as a 2.3 MB pdf file fromhttp://www.uh.edu/engines/vortexcylinders.pdf.[7.25] W.
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Heat transfer from tubes in crossflow. In T.F. Irvine,Jr. and J. P. Hartnett, editors, Advances in Heat Transfer, volume 8,pages 93–160. Academic Press, Inc., New York, 1972.[7.29] A. Žukauskas. Heat transfer from tubes in crossflow. In T. F.Irvine, Jr. and J. P. Hartnett, editors, Advances in Heat Transfer,volume 18, pages 87–159. Academic Press, Inc., New York, 1987.[7.30] S. Kalish and O. E. Dwyer. Heat transfer to NaK flowing throughunbaffled rod bundles.
Int. J. Heat Mass Transfer, 10:1533–1558,1967.[7.31] W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho, editors. Handbookof Heat Transfer. McGraw-Hill, New York, 3rd edition, 1998.8.Natural convection in singlephase fluids and during filmcondensationThere is a natural place for everything to seek, as:Heavy things go downward, fire upward, and rivers to the sea.The Anatomy of Melancholy, R. Burton, 16218.1ScopeThe remaining convection mechanisms that we deal with are to a largedegree gravity-driven. Unlike forced convection, in which the drivingforce is external to the fluid, these so-called natural convection processesare driven by body forces exerted directly within the fluid as the resultof heating or cooling. Two such mechanisms that are rather alike are:• Natural convection.
When we speak of natural convection withoutany qualifying words, we mean natural convection in a single-phasefluid.• Film condensation. This natural convection process has much incommon with single-phase natural convection.We therefore deal with both mechanisms in this chapter. The governing equations are developed side by side in two brief opening sections.Then each mechanism is developed independently in Sections 8.3 and8.4 and in Section 8.5, respectively.Chapter 9 deals with other natural convection heat transfer processesthat involve phase change—for example:397398Natural convection in single-phase fluids and during film condensation§8.2• Nucleate boiling.
This heat transfer process is highly disordered asopposed to the processes described in Chapter 8.• Film boiling. This is so similar to film condensation that it is usuallytreated by simply modifying film condensation predictions.• Dropwise condensation. This bears some similarity to nucleate boiling.8.2The nature of the problems of film condensationand of natural convectionDescriptionThe natural convection problem is sketched in its simplest form on theleft-hand side of Fig. 8.1. Here we see a vertical isothermal plate thatcools the fluid adjacent to it. The cooled fluid sinks downward to form ab.l.
The figure would be inverted if the plate were warmer than the fluidnext to it. Then the fluid would buoy upward.On the right-hand side of Fig. 8.1 is the corresponding film condensation problem in its simplest form. An isothermal vertical plate coolsan adjacent vapor, which condenses and forms a liquid film on the wall.1The film is normally very thin and it flows off, rather like a b.l., as thefigure suggests. While natural convection can carry fluid either upwardor downward, a condensate film can only move downward.
The temperature in the film rises from Tw at the cool wall to Tsat at the outer edgeof the film.In both problems, but particularly in film condensation, the b.l. andthe film are normally thin enough to accommodate the b.l. assumptions[recall the discussion following eqn. (6.13)]. A second idiosyncrasy ofboth problems is that δ and δt are closely related. In the condensingfilm they are equal, since the edge of the condensate film forms the edgeof both b.l.’s. In natural convection, δ and δt are approximately equalwhen Pr is on the order of unity or less, because all cooled (or heated)fluid must buoy downward (or upward). When Pr is large, the cooled (orheated) fluid will fall (or rise) and, although it is all very close to the wall,this fluid, with its high viscosity, will also drag unheated liquid with it.1It might instead condense into individual droplets, which roll of without forminginto a film. This process, called dropwise condensation, is dealt with in Section 9.9.§8.2The nature of the problems of film condensation and of natural convectionFigure 8.1 The convective boundary layers for natural convection and film condensation.
In both sketches, but particularly in that for film condensation, the y-coordinate has beenstretched.In this case, δ can exceed δt . We deal with cases for which δ δt in thesubsequent analysis.Governing equationsTo describe laminar film condensation and laminar natural convection,we must add a gravity term to the momentum equation. The dimensionsof the terms in the momentum equation should be examined before wedo this. Equation (6.13) can be written as∂umN1 dp m3∂ 2 u m2 m∂u+v=−+νu∂xs2∂yρ dx kg m2 · m∂y 2 s s · m2=kg·mkg·s2=Nkg=Nkg=ms2=Nkgwhere ∂p/∂x dp/dx in the b.l.
and where µ constant. Thus, everyterm in the equation has units of acceleration or (equivalently) force perunit mass. The component of gravity in the x-direction therefore enters399400Natural convection in single-phase fluids and during film condensation§8.2the momentum balance as (+g). This is because x and g point in thesame direction. Gravity would enter as −g if it acted opposite the xdirection.u∂u1 dp∂2u∂u+v=−+g+ν∂x∂yρ dx∂y 2(8.1)In the two problems at hand, the pressure gradient is the hydrostaticgradient outside the b.l. Thus,dp= ρ∞ gdxdp= ρg gdxnaturalconvectionfilmcondensation(8.2)where ρ∞ is the density of the undisturbed fluid and ρg (and ρf below)are the saturated vapor and liquid densities.
Equation (8.1) then becomes∂uρ∞∂2u∂u+v= 1−g+νfor natural convection(8.3)u∂x∂yρ∂y 2ρg∂u∂2u∂u+v= 1−for film condensation(8.4)g+νu∂x∂yρf∂y 2Two boundary conditions, which apply to both problems, are 1the no-slip conditionu y =0 =0 v y =0 =0no flow into the wall(8.5a)The third b.c.
is different for the film condensation and natural convection problems:⎫⎪∂u condensation:⎪⎪⎪=0⎬noshearattheedgeofthefilm∂y y=δ(8.5b)⎪⎪ ⎪natural convection:⎪⎭u y =δ =0undisturbed fluid outside the b.l.The energy equation for either of the two cases is eqn. (6.40):u∂T∂2T∂T+v=α∂x∂y∂y 2We leave the identification of the b.c.’s for temperature until later.The crucial thing we must recognize about the momentum equationat the moment is that it is coupled to the energy equation.
Let us considerhow that occurs:§8.3Laminar natural convection on a vertical isothermal surfaceIn natural convection: The velocity, u, is driven by buoyancy, which isreflected in the term (1 − ρ∞ /ρ)g in the momentum equation. Thedensity, ρ = ρ(T ), varies with T , so it is impossible to solve themomentum and energy equations independently of one another.In film condensation: The third boundary condition (8.5b) for the momentum equation involves the film thickness, δ. But to calculate δwe must make an energy balance on the film to find out how muchlatent heat—and thus how much condensate—it has absorbed.
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