Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 52
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4.106 is again recommended; when Null (Ra cos 0) is equated toNuv (Ra sin 0) to find the crossover angle 0~, this angle is generally found to be greater than60 °. Equation 4.106 agrees within about 10 percent with the data of Arnold et al. [9] for H/L =0.25, and Edwards et al. [80] for H/L -0.25 and H/L -0.14. No data on which to base scalinglaws seem to be available for 0.25 _<H/L <_1.NATURAL CONVECTION4.57Cavities with Pr = O. 7 and W/H > 8. For H / L > 5 and 0 _<0 < 60 °, direct application of thehorizontal scaling law (Eq. 4.104) introduces significant errors when Pr = 0.7 and Ra = 104[144]; these errors have been shown [66, 235] to result from a secondary instability thatappears at a Rayleigh number only slightly greater than that for the primary instabilitydiscussed in the section on horizontal rectangular parallelepiped and circular cylinder cavities.
A modified scaling relation, taken from Hollands et al. [144], is recommended for 0 _<0 < 60°:1708 1"[1708 (sin l "80)16 ] [ ( R a c ° s 0 ) 1/3 1"Nu0 (Ra) = 1 + 1.4411 - Ra cos 01Ra cos 0+5830- 1(4.110)This equation agrees well with data up to Ra = 105; for Ra = 106 it underestimates the measurements of E1Sherbiny et al. [84, 85] by 10 percent. (See Eq. 4.79 for meaning of dots.)For 0 = 60 ° the recommended relation, taken from E1Sherbiny et al. [84, 85], isNu60 (Ra) = [Nu 1, Nu2]max(0.0936 Ra°314)7 ]in1+ GwhereNu~0 = 1 +andNu~0 = (0.1044 + 0.1750 L ) Ra °283(4.111)0.5G = [1 + (Ra/3165)2°6] °1(4.112)(4.113)For 60 ° _<0 < 90 °, linear interpolation between the 60 ° and 90 ° relations is recommended:Nu0 (Ra) =90 ° - 00 - 60 °30-------:---Nu60 (Ra) + 300 Nuv (Ra)(4.114)The tested range of validity of Eqs. 4.111-4.114 is the same as that for Eq.
4.92.For 0.5 _<H / L <_5 and 45 ° < 0 _<90 °, Meyer et al. [191] recommend a relation that can be fitted byNu0 (Ra) = 1.06 Nuv (Ra)Nu0 (Ra) = 1 +0.06(90 ° - 0) ]300Nuv (Ra)45 ° < 0 < 60 °(4.115)60 ° < 0 < 90 °(4.116)For 3 < L / H <_10 and 0 < 0 _<75 °, Smart et al. [253] found that the horizontal scaling law (Eq.4.99) is valid. For ranges of 0 not covered in this section, the relevant scaling laws for Pr ~> 4are tentatively recommended (see the preceding section).Effect o f W l H on N u for Inclined Cavities. The data of Edwards et al. [80] indicate that forL / H = 4, horizontal scaling (which had been found valid for 0 _<0 _<60 ° when W / H = 24 and 8)becomes invalid for W/H = 4.
The data of Cane et al. [29], for which W / H = 1, agree well withthe following relation:N u = l + l ' 1 5 [ N u v ( R a ) - l l c ° s ( 0 - 6 0 ° ) ( R a H 4 164('-sin°)2840L)4(4.117)provided 0 _>30 ° and Ra H4/L 4 <_6000. Equation 4.117 was validated for L / H = 2, 3, 4, and 5.Plotted experimental data for a number of combinations of W/H, H/W, and 0 are given byEdwards et al. [80].4.58CHAPTER FOURHeat Transfer in Enclosures with Interior Solids at Prescribed TemperatureRegion betweenConcentric or Eccentric Cylinders.
The geometry and dimensions are asshown in cross section in Fig. 4.36a, the two cylinders being assumed to have parallel axes. Thedimension E represents the perpendicular distance from the axis of the inner cylinder to theaxis of the outer cylinder. Thus, for concentric cylinders, E = 0. Each cylinder is taken to beisothermal but at a different temperature.~o\L:Re :(Do-Di)/2g/~ (T~-To) L 3(a)(b)FIGURE 4.36 Sketch of concentric and eccentric cylinder and sphereproblems.The stages of flow development with increasing Ra parallel those in the vertical rectangular parallelepiped cavity.
For small Ra, the fluid flow (which is present for any finite Ra)is sufficiently feeble that the heat transfer is, for all practical purposes, by conduction only. AsRa is increased, a laminar boundary layer regime is established wherein the flow is largelyrestricted to boundary layers on each of the cylinders. The central region in the regime is stably stratified and almost stationary: it generally contains one or two large, slowly rotatingeddies on each side of the cavity, but the details of this flow structure have little effect on theheat transfer.
At high Ra the boundary layers can be expected to become turbulent.The Nusselt number for this problem is defined byNu =q' In (Do/Di)2x(T, - To)k(4.118)where q' is the heat transfer by conduction and convection from the inner cylinder to theouter one per unit axial length of cylinder. The temperatures Ti and To correspond to theinside diameter Di and the outside diameter Do. By NUCONDwe mean the Nusselt number thatapplies when the fluid is stationary and so there is no convection.
The definition for Nu prescribed by Eq. 4.118 makes NUCOND= 1 when the cylinders are concentric (E = 0), andIn (Do/Di)NUCOND= cosh_ 1 [(Do2 + 02i _ 4E2)/2OoOi ](4.119)when 0 < E < L.Figure 4.36b shows conduction layers applied to each cylinder.
Both conduction layerthicknesses are shown as being much less than the spacing L so that they do not touch--thissituation will always occur if Ra is large enough to make Ao + A/< L - E. Since, according tothe conduction layer model, the central region is isothermal, this model predicts that, provided the conduction layers do not touch, Nu will be independent of E. In fact, this is what isNATURAL CONVECTION4.59observed; for cylinders with vertical axes and for Ra = 4.8 × 10 4, Do/Di = 2.5, and Pr = 0.7,Kuehn and Goldstein [164] found that Nu changed by no more than 10 percent when E wasvaried from 0 to ~L, regardless of the direction of displacement of this inner cylinder.
(Calculations indicate that for this Ra, m i = m o ~ L/4, so a slight overlap of conduction layers wouldhave occurred at E = ~L.)Cylinders With Horizontal Axes. The conduction layer model has been shown to accurately predict the heat transfer for horizontal cylinders [164, 223], but because of the need toiteratively solve for the central region temperature it does not yield an explicit expression forNu.
However, by making additional approximations Raithby and Hollands [223] were able toderive an explicit relation for the heat transfer when the (assumed laminar) conduction layersdo not overlap:In (Do/Di) Ra TMNut = 0.603C, [(L/Di)3/5 + (L/Do)3/515/4--(4.120)For E = 0, the conduction layers just touch when Ra falls to the value at which Eq. 4.120 predicts Nut = NUCOND= 1; for still smaller values of Ra, Nu remains at NUCOND,SO that for theconduction and laminar flow regimesN u = [NUcoND , NUl]ma x(4.121)where Nut is given by Eq. 4.120 and NUCONDis equal to unity.
For E ;e 0, Eq. 4.121 can still beused, with Eq. 4.119 being used for NUCOND;however, since the conduction layers do not touchuniformly as Ra decreases toward the conduction regime, some error must be expectedaround Nut-- NUCOND.Well removed from this region, the equation should be accurate.Equation 4.121 was tested against earlier (i.e. prior to 1975) data with E = 0 over the ranges2 x 10z < Ra _<8 x 10 7, 0.7 < Pr _<6000, and 1.15 < Do/Di < 8; agreement was almost invariablywithin 10 percent. The more recent data of Kuehn and Goldstein [164] at Do/D~ = 2.6 and atPr =0.7, which covered the range 2 x 102 < Ra < 8 x 10 7, agree with Eq. 4.116 with a mean deviation of 2.1 percent and a maximum error, occurring when Nu = 1, of 8 percent.
The equationalso agrees very well with the numerical solutions of Farouk and Gtiqeri [96], which were forDo/D~ = 2.6 and ranged in Ra from 10 3 to 10 7. For Ra > 108, turbulence effects neglected in thederivation of Eq. 4.115 may become important, and the nonexplicit conduction layer model,expanded on by Kuehn and Goldstein [163, 164], should be adopted. Also, the equation maynot be accurate at very low Prandtl numbers. An equation with a wider Prandtl number rangehas been recommended by Hessami et al. [136] on the basis of their own data (which coveredthe ranges 0.023 < Pr < 10,000 and 1.15 < Do/Di < 11.4) as well as those of some other workers.Their equation isNu = 0.265[ln (Do/D~)][Ra ( O i / Z ) 3 ) ( l - D~/Do)/(1 + 0.952/Pr)] TM(4.122)Cylinders With Vertical Axes.
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