Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 42
Текст из файла (страница 42)
Using Tf = 0.5 Tw + 0.5 T.. as the reference temperature, instead of Te, raises the value ofC v by about 20 percent, to almost the same value as for water.The C v values for gases given by Eq. 4.24 assume that for AT/T~ << 1, or Tw/Too~ 1. If thiscondition is not met for a vertical plate, Eq. 4.21 should be rewritten asNut, x =Measurements by Pirovano et al.
[220, 221], Siebers et al. [251], and Clausing et al. [61, 63]suggest quite different values of f. Until further data become available, the recommendedequation for f in Eq. 4.30 is\ T. ]~- 1(4.31)The equations provided in this chapter assume that AT/T. ~ O. For large AT, the equation forC v should be replaced by fC v, where f is given by Eq. 4.31. It is concluded that there is considerable uncertainty in the recommended equation for C v for Pr > 1 and for large Tw/T..
Thisis perhaps the most important unresolved fundamental issue in the equations used to estimatenatural convection heat transfer.Discussion of C U Equation. Equation 4.25 for C,u has been forced to agree with the value0.15 measured by Lloyd and Moran [297] at Pr - 2000, and forced to pass through C,v = 0.14for air (Pr = 0.71) and water (Pr = 6). The latter values are deduced from measurements ofheat transfer across horizontal fluid layers (see the section on natural convection within4.12CHAPTERFOURenclosures) that, for Ra ~ 0% are correlated by Nu = c Ral/3; in this limit the relation betweenc and Cff is CY = c 2 4/3.
C U -- 0.14 for air and water agrees to within experimental scatter withmeasurements of heat transfer from horizontal upward-facing plates obtained by Yousef et al.[290], Bovy and Woelk [23], Clausing and Berton [62], Grober et al. [119], Hassan andMohamed [123], Fishenden and Saunders [98], A1-Arabi and E1-Riedy [4], Weiss [282], andFujii and Imura [103]. The single data point for mercury (Pr = 0.024), deduced from the horizontal layer measurements of Globe and Dropkin [114], is C~ = 0.13, which compares wellwith 0.14 from Eq. 4.25.Blending of Laminar and Turbulent Nusselt Numbers.The previous two sections providedheat transfer equations for the cases where there is laminar heat transfer from the entire body,(Nue), and turbulent heat transfer from the entire body, (Nut).
To obtain a fit to heat transferdata over the entire range of Ra, the blending equation of Churchill and Usagi [54] is used:N H = (NH~n ~- N H ? ) 1/m(4.32)The appropriate value of m, which generally lies in the range from 4 to 20, is chosen so as togive the best fit to the experimental data. Where no data for the body shape at hand are available, the value for a known body of similar shape is tentatively recommended; Tables 4.2(found on page 4.24) and 4.3 (found on pages 4.26-4.31) give values of m for a range ofshapes. Minor modifications to this prescription are sometimes implemented in order to moreaccurately represent the dependence of heat transfer on Ra through the transition.The aim of the correlation method has been to give an equation for Nu (Ra) that coversthe entire range of Ra from zero to infinity. On the other hand, the Rayleigh number range ofinterest in the particular problem at hand may be low enough to ensure that laminar heattransfer dominates.
In this case it may be preferable to simplify the equation by assuming thatthe turbulent term in Eq. 4.32 does not contribute appreciably, and take Nu as being equal toNu~. This is allowable if, at the highest Rayleigh number of interest, the difference betweenthe calculated values of Nu and Nue is acceptably small.Although certain of the steps in the development of the Nu (Ra) function given in this section have assumed that the body is convex, in practice, a minor degree of concavity has notbeen found to compromise its usefulness. For example, it has been applied to long cylinderswith an "apple core" cross section and to 3D bodies in the shape of an apple core.
A test is toexamine the degree to which the conduction layer on one part of the surface overlaps withthat on another part; if the degree of overlap is modest, the method should be useful. Anextreme case of concavity is the casem of a body with a vertical hole passing through it. Provided the conduction layer thickness A (Eq. 4.16) is thinner than, say, about 1~ the hole radius,the method should continue to work satisfactorily. If not, the hole should be treated separately as an open cavity, using methods explained in the section on open cavity problems.EXTERNAL NATURAL CONVECTIONEquations are presented in this section for evaluating the heat transfer by natural convectionfrom the external surfaces of bodies of various shapes. The correlation equations are of theform described in the section on the heat transfer correlation method, and the orientation ofthe surface is given by the surface angle ~ defined in Fig.
4.4. Supporting experimental evidence for each such equation set is outlined after each equation tabulation. The correlationsare in terms of Nu, Ra, and Pr, parameters that involve physical properties, a length scale, anda reference temperature difference. Rules for the evaluation of property values are providedin the nomenclature, and the relevant length scale and reference temperature difference areprovided in a separate definition sketch for each problem.NATURAL CONVECTION4. ] 3Flat PlatesW --------~Vertical Flat Plate with Uniform T , and T=, ~ = 9 0 °Correlation. In terms of quantities defined in Fig. 4.6Twand the nomenclature, the total heat transfer from a wide(W >> L) vertical isothermal plate can be estimated from thefollowing equations:LmxT~Lq~mgl3~TL3Nu=~Nux=~g~q"L4RaRa*-Ra;-v~k-RaNuRax-VO~g~q"x'v~kgl3z%Tx3Nur = Ct Ra TM(4.33a)2.0Nut = In (1 + 2.0/Nu r)(4.33b)Nu, = C v Ral/3/(1 + 1.4 x 109 Pr/Ra)(4.33c)Nu = [(Nut) m + (Nu,)m] 1/m(4.33d)VO~m=6- RaxNuxFor large AT/T, Cv should be replaced by C~Vf,where fis givenby Eq.
4.31.Comparison With Data. Figure 4.7 compares Eq. 4.33 tofour sets of measurements for gases for small AT/T. Therange of experimental data is 10-1 < Ra < 1012, and the RMSdeviation of the data from the equation is 5 percent. The data of Saunders [239] have beenreevaluated using more accurate property values. The data of Pirovano et al.
[220, 221] are fortheir lowest Tw/T= ratio, and only the laminar data of Clausing and Kempka [63] are shown.Equation 4.33 also closely fits the data for other fluids compiled by Churchill [56].Outstanding Issues. There are no data at low Ra for Pr < 0.7. There is also uncertaintysurrounding the expression for Ctv (Eq. 4.24) for both Pr << 1 and Pr >> 1. The transition inEq.
4.33 is assumed to depend exclusively on Ra/Pr; in fact, the transition appears to alsodepend on the vertical stratification within the ambient medium and on the size of AT.FIGURE 46 Definition sketch for natural convection on a vertical plate with uniform wall temperatureor uniform heat flux.lO3io Saunders[239]a~•Wamer&Arpaci [277]J•Pirovanoet al [220 221]/t~ Clausing&Kempka[63]102tt/tm101100•10-2I100,I102iI~104I106,I108,I101°,1012RaFIGURE 4.7 Comparisonof Eq. 4.33 with data for an isothermal, vertical flat plate in air.4.14CHAPTERFOURVertical Plate With Uniform Surface Heat Flux and Constant Too.
When the surface heatflux, q", is uniform and known, values of Nux and Nu are used respectively t_oocompute thelocal temperature difference AT and the average temperature difference AT. Parametersrelated to this problem are defined in Fig. 4.6 and in the nomenclature.Calculation of Local AT. The local AT can be found from the value of Nux calculatedfrom the following equation for Ra* > 0.1.Nuxr = He(Ra*) 1/5(He =Pr)1/5(4.34a)4 + 9~/~r + 10PrNue, = 0.4/ln (1 + 0.4/Nu~r)(4.34b)(CV)3/4(Ra*) TMNu,,x = 1 + (C2 Pr/Ra*) 310 ~2< C2 < 2 × 10 ~3C2 = 7 × 10 ~2(nominal)Nux = ( ( N u e j m + (Nu,,x)m)'/m(4.34c)m = 3.0(4.34d)The value of C2 determines the transition between laminar and turbulent heat transfer. Anominal value is C2 = 7 × 10 ~2.Comparison With Data. Equation 4.34 is in excellent agreement with the data of Humphreys and Welty [147] and Chang and Akins [40] for Pr = 0.023 (mercury), but the data lieonly in the laminar regime.
There is also good agreement with measurements of Goldsteinand Eckert [115], Vliet and Liu [275], and Qureshi and Gebhart [222] for water, although theobserved transition depends on the level of heat flux. By choosing an appropriate value for C2for each q", Fig. 4.8 shows that Eq.
4.34 can be made to fit each data set. For a nominal value103-"'......i.s.s3E2Cz1.06r131.32 E31.04 E13F12.33E38.01 E l 20371E35 97 E12•449E333s 12q 'tW/m 2]©./X,",,"'"'," '•I•- " '';"=e.Z..~, -o/~f..:~,~- , ~ , ~ : : ~///-v,,6:a;~'/-../ .."" /xzf210012.',.....!!10141013. . . . .1015Ra* xFIGURE 4.8 Comparison of local heat transfer measurements of Qureshi and Gebhart[222] with Eq.
4.34 for a uniform heat flux, vertical flat plate in water.NATURAL CONVECTION4.15by C2 = 7 x 1012, Eq. 4.34 gives good agreement with measurements for oils, Pr = 60 and Pr =140, except in the transition region. Fujii and Fujii [99] have shown that strong vertical temperature gradient in_the ambient fluid also affects the value of Nux in the transition regime.Calculation of AT. In some cases involving uniform q", it is sufficient to know the average temperature difference, defined asAT= -A(Tw- T=) dA(4.35)This is obtained from the average Nusselt number Nu defined in Fig. 4.6. A rough estimate ofNu can be obtained for Ra > 105 by using the equations in the section on vertical flat plateswith uniform Tw and T=, ~ = 90 °, with AT replaced by AT.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
