Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 84
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The data obtained by Vliet (1969) for a uniformflux heated surface in air and in water indicate the validity of this procedure up toinclination angles as large as 60°. Additional experiments have confirmed that thereplacement of g by g cos γ in the Grashof number is appropriate for inclinationangles up to around 45° and, to a close approximation, up to a maximum angle of60°. Detailed experimental results on this problem were obtained by Fujii and Imura(1972).
They also discuss the separation of the boundary layer for the inclined surfacefacing upward.The natural convection flow over horizontal surfaces is of considerable importancein a variety of applications, for instance, in the cooling of electronic systems andin flows over the ground and water surfaces. Rotem and Claassen (1969) obtainedsolutions to the boundary layer equations for flow over a semi-infinite isothermalhorizontal surface. Various values of Pr, including the extreme cases of very largeand small Pr, were treated. Experimental results indicated the existence of a boundarylayer near the leading edge on the upper side of a heated horizontal surface.
Theseboundary layer flows merge near the middle of the surface to generate a wake orplume that rises above the surface. Equations were presented for the power law case,Tw − T∞ = Nx n , and solved for the isothermal case, n = 0. Pera and Gebhart (1972)have considered flow over surfaces slightly inclined from the horizontal.For a semi-infinite horizontal surface with a single leading edge, as shown in Fig.7.6, the dynamic or motion pressure pd drives the flow.
Physically, the upper side of aheated surface heats up the fluid adjacent to it. This fluid becomes lighter than the ambient, if it expands on heating, and rises. This results in a pressure difference, whichcauses a boundary layer flow over the surface near the leading edge. Similar considerations apply for the lower side of a cooled surface. The governing equations are thecontinuity and energy equations (7.15) and (7.17) and the momentum equationsu∂u1 ∂∂u+v=∂x∂yρ ∂ygβ(T − T∞ ) =1 ∂pdρ ∂yµ∂u∂y−1 ∂pdρ ∂x(7.40)(7.41)This problem may be solved by similarity analysis, as discussed earlier for verticalsurfaces.
The similarity variables, given by Pera and Gebhart (1972), areBOOKCOMP, Inc. — John Wiley & Sons / Page 541 / 2nd Proofs / Heat Transfer Handbook / Bejan[541], (17)Lines: 522 to 553———4.21005pt PgVar———Normal PagePgEnds: TEX[541], (17)542123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONFigure 7.6 Natural convection boundary layer flow over a semi-infinite horizontal surface,with the heated surface facing upward.yη=xGr x51/5Gr xψ = 5νf(η)51/5(7.42)Figure 7.7 shows the computed velocity and temperature profiles for flow over aheated horizontal surface facing upward or a cooled surface facing downward. For aheated surface facing downward or a cooled surface facing upward, a boundary layertype of flow is not obtained for a fluid that expands on heating.
This is because thefluid does not flow away from the surface due to buoyancy. The local Nusselt numberfor horizontal surfaces is given by Pera and Gebhart (1972) for both the isothermaland the uniform-heat-flux surface conditions. The Nusselt number was found to beapproximately proportional to Pr 1/4 over the Pr range 0.1 to 100. The expression givenfor an isothermal surface isNux =hx x1/4= 0.394Gr 1/5x · Prk(7.43)and that for a uniform-flux surface, Nux,q , isNux,q =hx x1/4= 0.501Gr 1/5x · Prk(7.44)It can be shown by integrating over the surface that for the isothermal surface, theaverage Nusselt number is 53 times the value of the local Nusselt number at x = L.Therefore, the natural convection heat transfer from inclined surfaces can betreated in terms of small inclinations from the vertical and horizontal positions, detailed results on which are available.
For intermediate values of γ, an interpolationbetween these two regimes may be used to determine the resulting heat transfer rate.Numerical methods, such as the finite difference method, can also be used to solvethe governing equations to obtain the flow and temperature distributions and the heattransfer rate. This regime has not received as much attention as the horizontal andBOOKCOMP, Inc.
— John Wiley & Sons / Page 542 / 2nd Proofs / Heat Transfer Handbook / Bejan[542], (18)Lines: 553 to 579———0.4231pt PgVar———Normal PagePgEnds: TEX[542], (18)EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES123456789101112131415161718192021222324252627282930313233343536373839404142434445543[543], (19)Lines: 579 to 593———0.12099pt PgVarFigure 7.7 Calculated (a) velocity and (b) temperature distribution in natural convectionboundary layer flow over a horizontal surface with a uniform heat flux.
(From Pera and Gebhart,1972.)———Normal PagePgEnds: TEXvertical surfaces, although some numerical and experimental results are availablesuch as those of Fujii and Imura (1972).[543], (19)7.4 EXTERNAL LAMINAR NATURAL CONVECTION FLOWIN OTHER CIRCUMSTANCES7.4.1Horizontal Cylinder and SphereMuch of the information on laminar natural convection over heated surfaces, discussed in Section 7.3, has been obtained through similarity analysis. However, neitherthe horizontal cylindrical nor the spherical configuration gives rise to similarity, andtherefore several other methods have been employed for obtaining a solution to thegoverning equations.
Among the earliest detailed studies was that by Merk and Prins(1953–54), who employed integral methods with the velocity and thermal boundarylayer thicknesses assumed to be equal. The variation of the local Nusselt numberwith φ, the angular position from the lower stagnation point φ = 0°, is shown in Fig.7.8 for a horizontal cylinder and also for a sphere. The local Nusselt number Nuφdecreases downstream due to the increase in the boundary layer thickness, which ispredicted to be infinite at φ = 180°, resulting in a zero value for Nuφ there. However,Merk and Prins (1953–54) indicated the inapplicability of the analysis for φ ≥ 165°due to boundary layer separation and realignment into a plume flow near the top.BOOKCOMP, Inc. — John Wiley & Sons / Page 543 / 2nd Proofs / Heat Transfer Handbook / Bejan5440.7Pr = ⬁10Nu /(Gr Pr)1/40.60.50.41.00.30.70.20.100306090 (deg)120150180[544], (20)(a)0.8Pr = ⬁0.7Nu /(Gr Pr)1/4123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONLines: 593 to 605———100.14706pt PgVar———Long PagePgEnds: TEX0.60.51.00.40.70.3[544], (20)0.20.100306090 (deg)120150180(b)Figure 7.8 Variation of the local Nusselt number with downstream angular position φ for(a) a horizontal cylinder and (b) a sphere.
(From Merk and Prins, 1953–54.)The mean value of the Nusselt number Nu is given by Merk and Prins (1953–54)for a horizontal isothermal cylinder asNu =h̄D= C(Pr)(Gr · Pr)1/4k(7.45)where Nu and Gr are based on the diameter D. The constant C(Pr) was calculated as0.436, 0.456, 0.520, 0.523, and 0.523 for Pr values of 0.7, 1.0, 10.0, 100.0, and ∞,respectively.BOOKCOMP, Inc.
— John Wiley & Sons / Page 544 / 2nd Proofs / Heat Transfer Handbook / BejanEXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES123456789101112131415161718192021222324252627282930313233343536373839404142434445545The preceding expression is also suggested for spheres by Merk and Prins (1953–54), with C(Pr) given for the Pr values of 0.7, 1.0, 10.0, 100.0, and ∞ as 0.474,0.497, 0.576, 0.592, and 0.595, respectively. There are many other analytical andexperimental studies of the natural convection flow over spheres.
Since this configuration is of particular interest in chemical processes, it has also been studied in detailfor mass transfer. Chiang et al. (1964) solved the governing equations, using a seriesmethod, and presented heat transfer results. Trends similar to those discussed earlierwere obtained.
A considerable amount of experimental work has been done on theheat transfer from spheres. Amato and Tien (1972) have discussed such studies andhave given the heat transfer correlation asNu = 2 + 0.5(Gr · Pr)1/4(7.46)where the constant 2 in the expression can be shown analytically to apply for pure conduction. Additional correlations for transport from horizontal cylinders and spheresare given later. Work has also been done on the separation of the flow to form awake near the top of the body.
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