Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 47
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This results from the rapid increasein total surface area with decreasing fin spacing (i.e., as fins are added to the base plate) forhigh fins. Because the total heat transfer falls off very sharply for spacings below the optimum, a conservative design would use spacings larger than the optimum $2 (defined in thesection on rectangular isothermal fins on vertical surfaces) calculated from Eq. 4.62.Horizontal Corrugated Surfaces. A heated horizontal isothermal corrugated surface, suchas that shown in Fig. 4.23c, can also be considered a finned surface. The heat transfer is fromthe top surface only. AI-Arabi and EI-Refaee [3] have measured heat transfer rates to air(Pr =0.71) over the range 1.8 x 104 < Ra < 1.4 x 10 7 (the nomenclature is defined in Fig.
4.23c),and have provided the following correlations:(0.46)Nu = sin (W/2) -0.32 Ra mwherefor 1.8104 < Ra < Rac(4.63)for Ra~ < Ra < 1.4 x 107(4.64)×Rac = (15.8- 14.0 sin (W/2)) x 105m = 0.148 sin (re/2) + 0.187and054) Ral/3Nu = ( 0.090 + sin0 .
(~/2------~The measurements and the above correlation were intended to represent the case where W isasymptotically large. The dependence of the heat transfer on W remains to be determined.Vertical Triangular Fins. For the vertical fin array in Fig.
4.23d, S is the fin spacing measured at the mid-height of the fin, so that SW is the cross-sectional area of the flow channelformed by the sides of adjacent fins, the base of width S, and the vertical plane passing4.38CHAPTER FOURthrough the fin tips. Nu and Ra are defined in the figure. The correlation of Karagiozis et al.[152] isNu = NUcoND + C~ Ra 1/4 1 +RaO2~+ 8 Nu8 Nu = [(0.147 Ra °39- 0.158 Ra°46), 0]max(4.65a)(4.65b)where NUCOND is the (conduction) Nusselt number for the entire fin array in the limit asRa -4 0.
A conservative design (i.e., the heat transfer is underestimated) results if NUcoND = 0is used.Otherwise, to estimate NUCOND,suppose that triangular fins are mounted on a rectangularbase plate of dimension L1 and H, where the base plate entirely covers the vertical surface onwhich it is mounted. The surface and base plate therefore have a r e a A b = L1 × H. Supposefurther that the fin height is small compared to the base plate dimensions (i.e., W << L1 andW << H), and that there are no adjacent cool surfaces to which heat can be directly conducted.The lower bound on NUCOND(see Refs.
293 and 294) can be found fromNUcoND =~(1 + ~ r ) 2X/-~br_=S ~r<5(4.66a)r>5(4.66b)X/~rIn (4r)where r = (L1/H, H/L1) . . . . The upper bound for NUcoND is roughly 1.57 times the value calculated from Eq. 4.66; this factor is obtained by extrapolating the recommendation ofYovanovich and Jafarpur [293], intended for r -- 1. In the absence of better information, usethe average of these two values.If the base plate of the fins is attached to a vertical isothermal surface whose dimensionsare much larger than both H and L1, NUcoND will approach zero. If, however, there is a nearbycool surface to which heat can be transferred, NUcoND can be larger than the upper bound justdescribed; this effect becomes important when the nearest distance to the cool surface is lessthan S / ( N u - NUCoND) and where ( N u - NUcoND) is obtained from Eq.
4.65.Figure 4.24 compares Eq. 4.65, shown by the solid line, to the data of Karagiozis et al. [152,153]. The data approach the asymptote for laminar heat transfer from a vertical flat plate(dotted line) at high Ra. The dashed line shows Eq. 4.65a with no correction (i.e., 8 Nu = 0);this correction is seen to be significant only at low Ra.Finned surfaces are often installed on a vertical surface with the fin tips running horizontally (instead of vertically).
Karagiozis [153] has shown that this reduces the heat transfer byup to a factor of 2, and the use of this orientation is not recommended.Square Isothermal Fins on a Horizontal Tube. Square fins attached to a horizontal tube, asshown in Fig. 4.23e, are commonly used in heat exchangers. The experimental data of Elenbaas [88, 89] for square plates, without the tube, covered the range 0.2 < Ra < 4 × 104, for Pr =0.71. These data are closely correlated byNu =l/Ra089)m18+ (0.62 Ral/4)m]l/mm = -2.7(4.67)Recent heat transfer measurements by Sparrow and Bahrami [256, 257] lie about a factor of10 higher than those of Elenbaas near the lower end of the Ra range. Tsubouchi and Masuda[269] recommended that the heat transfer from square fins be calculated from their equationsfor circular fins using an equivalent diameter D = 1.23H.
Their procedure for circular fins isoutlined in the following section.NATURAL CONVECTION4.39101 ~.'~5H',III"_+10O/o..'"--"""iz/~zW/So 6.67.~-HIS20.0S/S0.010-1 _".... Flat plate//10-2S "10-2tIII10 °102104108RaFIGURE 4.24 Comparison of Eq.
4.65 with data of Karagiozis et al. [152] for vertical isothermal triangular fins.Circular Isothermal Fins on a H o r i z o n t a l Tube. Tsubouchi and Masuda [269] measuredthe heat transfer by natural convection in air from circular fins attached to circular tubes, asin the configuration shown in Fig. 4.23f Correlations for the heat transfer from the tips of thefins (see the figure for definition), and from the cylinder plus vertical fin surfaces, werereported separately.The average heat transfer from the tips was correlated byNu = c Ra b(4.68)where Nu and Ra are defined in Fig.
4.23f Data were obtained for 2 < Ra < 104, and c and bare listed in Table 4.5 for various values of fin-to-cylinder diameter D/d. For 1.36 < D/d < 3.73,the following approximations can be used: b = 0.29, c = 0.44 + 0.12D/d.The heat transfer from the lateral fin surfaces together with the supporting cylinder werecorrelated [269] for high fins, 1.67 < D/d < oo, byRaNu = 1 - ~TABLE 4.5{2-[ // C1 ~3/4]]]exp[-(~C-~l] 3/4] exp[-I]/-~-a)L \Ka/j(4.69a)Values of c and b Calculating the Heat Loss From Fin Tips (Eq.
4.68)D/dcb3.733.002.451.821.361.140.90.290.80.290.660.290.660.290.620.290.590.274.40CHAPTER FOUR13= 0.17~ + e-48;;where~ = d/DC1= [ 23"7 -1"1(1+152~2)1/2[3](4.69b)Properties are based on the wall temperature. This equation is in excellent agreement withTsubouchi and Masuda's data over the measurement range 3 < Ra < 104.For shorter fins, 1.67 > D/d > 1.0, Eq. 4.69a is replaced byC1c2c3Nu = C0 Ra~ { 1 - exp[-(-~---~ ) ] }whereCo = -0.15 + 0.3~ + 0.32~ 16C 1 ---180 + 480~ - 1.4~-8C2 = 0.04 + 0.9~C 3 --1.3(1 - ~) + 0.0017~ -12p = V'4 + CaC3Ra0 = Ra/~ = Ra(4.70a)(4.70b)D/dProperties are again evaluated at the wall temperature.Edwards and Chaddock [81] correlated their data for heat transfer from the entire surface,including the tip, for D/d = 1.94, 5 < Ra < 104, by= 0 las.aO1°294(4.71)where properties are evaluated at Too+ 0.62(Tw + Too).The measurements of Jones and Nwizu[151] fall slightly below Eq.
4.71. The fact that these equations do not have the expected fullydeveloped behavior (Nu o~ Ra) as Ra ---) 0 has been attributed [269] to tip effects.NATURAL CONVECTION WITHIN ENCLOSURESIntroductionEnclosure problems (Fig. 4.1c) arise when a solid surface completely envelops a cavity containing a fluid and, possibly, interior solids. This section is concerned with heat transfer by natural convection within such enclosures.
Problems without interior solids include the heattransfer between the various surfaces of a rectangular cavity or a cylindrical cavity. Theseproblems, along with problems with interior solids including heat transfer between concentricor eccentric cylinders and spheres and enclosures with partitions, are discussed in the following sections. Property values (including 13) in this section are to be taken at Tm = (Th + Tc)/2.Geometry and List of Parameters for Cavities Without Interior SolidsThe problem of natural convection in a cavity without interior solids is exemplified by the twosituations sketched in Fig.
4.25. In both situations, the fluid-filled cavity is bounded by twoisothermal parallel "plates" that are inclined at angle 0 from horizontal, spaced at distance L,and held at different temperatures. The temperature Th is assumed to be larger than To socavities with 0 = 0 ° are described as horizontal with heating from below, those with 0 = 90 ° aredescribed as vertical with heating from the side, and those with 0 = 180 ° are described as horizontal with heating from above.NATURAL CONVECTION~ ~ U p p e r4.41ploteetTc(a)(b)FIGURE 4.25 The (a) rectangular parallelepiped and (b) truncated circular cylinder cavities. Angle 0 is measured from the horizontal.In Fig.
4.25a the cavity is a rectangular parallelepiped, in Fig. 4.25b a truncated circularcylinder; the area over which the fluid contacts each plate is A i = H W for the first, and A i =riD2~4 for the second. The "wall" (as distinct from the plates) bounding the fluid on the sidesis of uniform thickness b and thermal conductivity kw. T h e upper and lower plates extend overthe wall as shown, and each therefore contacts the wall over an area Aw = (2b + H)(2b + W) H W for the first shape, and Aw = (n/4)[(D + 2b) 2 - D 2] for the second. The thermal boundarycondition on the outside faces of the wall (i.e., those faces not bounded by either the fluid orthe plates) is adiabatic.
The fluid is assumed to be either completely transparent or completely opaque to thermal radiation. Usually liquids are opaque and gases transparent. Whenthe fluid is transparent, radiant exchange can affect the free convection by altering the walltemperature distribution: hemispherical emissivities eh, ec, and ew are assigned to the surfacesof the hot plate, the cold plate, and the walls, respectively.Two different Nusselt numbers can be assigned for both cavity problems.
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