Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 44
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4.44 for Nut into Eq. 4.33d toFIGURE 4.14 Definitionsketch for natural convec- obtain Nu. For an isoflux boundary, calculate Nut from Eq.tion from long vertical cylinders.4.36c, calculate Nut from Eq. 4.44, and find Nu by substituting these values of Nu, and Nut into Eq. 4.36d.For a cylinder of noncircular convex cross section, such as the triangular cylinder shownin Fig.
4.14, follow the same procedure but replace D in Eq. 4.44 by Deft = ei/g, where Pi isthe perimeter of the cylinder.Discussion. This procedure is compared to available measurements in Fig. 4.15. The predictions fall about 10 percent higher than the measurements of Nagendra et al. [203] for a uniform flux into water. There is excellent agreement with the measurements of Li and Tarasuk[180] for 14 < L/D < 21 for a uniform wall temperature in air, and predictions fall slightlybelow the measurements of A1-Arabi and Salman [6], also for air.
There is also good agreement with the data of Kyte et al. [172] for a vertical wire of extremely small diameter (D = 0.08mm, L/D = 5430) in air where the transverse curvature correction in Eq. 4.44 is very large (themeasured Nu values in Fig. 4.15 fall below the predictions at the low end of the Nu rangebecause of rarefied gas effects).Ra*-v~xk3000.........., , - - / , 120001000 ~-o AI-Arabi & Salman, 2<1JD<16, air~-/•Li & Tarasuk, 14<L/D<21, air600zx Nagendra et al, 200<1./13<500,w a t e r400El Kyteet/ //c-.E_300Q.
200XWz100sol604030201.5,20,,3040, o . ,60oulO0,,,200300400Nu Predicted,600,1000,20003000FIGURE 4.15 Comparisonof recommendedprocedure for calculating heat transfer todata for vertical cylinders in air and water.4.21NATURAL CONVECTIONLdT** /A-TS = D - - ~zNu-~~"Dq"DATkN u - A'l'kRa =gl3ATD3Ra =.gl3ATD3./VO~..q"DN u - ATk."A"Tgl3ATD3Ra=~vo~._I~Tw 71"VO~(a)jp = 2 --6-cot 00(c)(b)F I G U R E 4.16 Definition sketch for long horizontal cylinder in an isothermal medium (a), in a stratified medium (b), and for atilted cylinder (.c).The present recommendation leads to good agreement with existing data, but none ofthese data lie in the turbulent range, and the tested range of Pr and L / D is also limited.A correlation for short vertical cylinders, including heat transfer from the ends, is providedin the section on bodies with small aspect ratios.Long H o r i z o n t a lCircular CylindersFor a long isothermal horizontal circular cylinder in an isothermal environment, and using the nomenclature in Fig.
4.16, Nu is obtained from the following equations:Correlation.Nur = 0.772Ce R a(4.45a)TMYNut = In (1 + 2f/Nu r)0.13f = 1 - [NurX°'l-'~t)Nut = Ct Ra v3Nu = [(Nue) m + (NUt)m] 1/m(4.45b)(4.45c)m = 10(4.45d)Ct can be found from Table 4.2 (elliptical cylinders, C/L = 1.0). If Ra > 10-4, the expression forf i n Eq. 4.45b can be replaced by f = 0.8.Discussion.
Equation 4.45 follows the recommendation of Raithby and Hollands [227]except that the value of m has been improved. There is a large body of data for this problem,because the shape is so frequently encountered in practice. Five sets of data, thought to beespecially reliable, are compared to Eq. 4.45 in Fig. 4.17 for air. The RMS deviation from theequation is less than 5 percent for 10-1° < Ra < 10 7. Similar agreement is found with the dataof Chen and Eichhorn [42] for water and with the data of Sch__~tz [300] for Pr = 2000.
HighRayleigh number data are still needed to confirm the value of C, in Eq. 4.45c and Table 4.2.For a uniform heat flux condition, apply Eq. 4.45 to obtain the average temperature difference AT.Stratified Medium. For a long horizontal circular cylinder in a thermally stratified environment in which the temperature increases linearly with height (see Fig. 4.16b for nomenclature), AT is the temperature difference at the mid-height of the cylinder. First calculate thelaminar isothermal Nusselt number Nue from Eq. 4.45 with AT = AT and rename it Nue,iso;the corresponding calculated heat flow is qiso. The laminar Nusselt number Nue corrected toaccount for the stratification is then estimated fromq__q_ Nue _ 1 +qisoS _<2(4.46a)S _>2(4.46b)Nue,iso= cS TMFor gases: a = 1, b - 0.39, and c = 1.29.
For water (Pr = 6): a = 2, b = 0.53, and c - 1.21.4.22CHAPTER FOUR102'••''I....I"'•'"'f""'o Clemesel al. [64]• DeSocio[71]#•'~//Hesse and Sparrow [137]101ALi andTarasuk [180]vCollisandWilliams [ 6 7 ]j~~J"TZm.10010-1 -lO . . . . . .10'10 -5•I,10 0,,,.L10 5RaFIGURE 4.17 Comparisonof Eq. 4.45 to data for long, horizontal, isothermal cylindersin air.Discussion of Stratified Medium.
Equation 4.46 was obtained by fitting the analyticalresults of Chen and Eichhorn [42]. There is agreement with their measurements of Nu/Nuisoto within about 10 percent. All their data were obtained for 2 x 105 < Ra < 3 x 107. Use of Eq.4.46 is not recommended for Ra much outside this range, because turbulent heat transfer willnot be properly accounted for at larger Ra and because thick boundary layer effects may bepoorly approximated for smaller Ra.Long Inclined Circular Cylinder, Laminar Flow.See Fig. 4.16c for nomenclature.
Thecylinder axis is inclined from the horizontal at an angle 0. The heat transfer from the ends isignored, which will be valid for insulated ends or for L/D > 5. The analysis of Raithby andHollands [225,226] led to the following correlation:Nur = ( 0.772 + 1 + 0.676p| 4Ce Ra TM1230"228 } ( cos 0 + -L-Dsin 0 /W(4.47a)1.8Nue = In (1 + 1.8/Nu r)(4.47b)LP = 2 -~- cot 0Equation 4.47a can be shown to reduce exactly to Eqs. 4.33a and 4.45a for the vertical (0 =90 °) and horizontal (0 = 0 °) orientation, respectively.Discussion. Equation 4.47 agrees with the data of Oosthuizen [213] and Li and Tarasuk[180] to within RMS and maximum errors of 6 and 12 percent, respectively. The data range is2.0 x 104 < Ra < 2.5 x 105, Pr = 0.71, 8 < LD < 21, and 0 ° < 0 < 90 °, and the fluid is air.
Figure4.18 presents a comparison for one case, and shows that the heat transfer falls as the cylinderis rotated from the horizontal (0 = 0 °) to the vertical (0 = 90 °) orientation.NATURAL CONVECTION0.55...o0.5....4.23.Oosthuizen [213]0 . 4 5 --d0.4-n-Pr=0.71Ra=26980z 0.35 -IJD=8.000.3-0.25 -0"2oi'02'04'03'06'0Angle from the Horizontal, 07'08'090FIGURE 4.18 Dependence of heat transfer on the inclination angle of a cylinder(defined in Fig.
4.16c): comparison of Eq. 4.47 with the data of Oosthuizen [213].Long Horizontal Noncircular CylindersCorrelation for Specific Shapes. For the four shapes shown in Table 4.2, the heat transferis calculated from the following equation:G-CeRa TM(4.48a)C2Nue = In (1 + C2/Nur)(4.48b)Nut = Ct Ral/3(4.48c)Nu = [(Nue) m + (Nu,)m] ~'m(4.48d)Nu r =The constants G, C2, -C,, and m are tabulated in Table 4.2, which also gives the definitions ofNu and Ra for each cylinder shape in addition to the equation for the perimeter P that isrequired to find the surface area.Discussion of Correlation for Specific Shapes.
The equation for elliptical cylinders fits theapproximate analysis of Raithby and Hollands [224] but has not been verified by experimentexcept in the limiting cases of a vertical plate (C/L = 0) and a circular cylinder (C/L = 1.0). Thevertical plate predictions by Eq. 4.48 are slightly different than those based on the more accurate specialized equations given in the section on external natural convection in flat plates.Equation 4.48 agrees with the measurements by Clemes et al.
[64] for square cylinders atangles of rotation of 0 °, 15 °, 30 °, and 45 ° in air (Pr = 0.71) for 101 < Ra < 107 to an RMS deviation of less than 3 percent and a maximum error of 8 percent. Similar comparisons for thesemicircular cylinder at 0 = 0 °, 45 °, 90 °, -45 °, a n d - 9 0 ° give RMS and maximum deviationsfrom the measurements of Clemes et al. [64] of 2.5 and 5 percent, respectively. For the slottedcylinder, the RMS and maximum deviations were 2 and 5 percent, respectively.r-,r~006-04.240§0000~0....~.~ai~.~ ~vv~ ~~'~Vv0 0 0 0II0=a~~*"IIII~~C-~II0 0 0 0 00 0 0 0 00 0 0 0 0~ZIIo 0 0 00 0 0 00 0 0 00 0 0 0~D0 0 0 0 0II"~~ o oZII0 0 0 0IIII:~Z0 0 0 0 0 0 0II0 0 0 0 0~0 0 0 0IId d d d d d dZ0II~3<'~IIVNATURAL CONVECTIONeter, PT,zf --ff4.25There appears to be no verification of Eq. 4.48 for theseshapes for fluids other than air, and for Rayleigh numbers(defined in Table 4.2) outside the range 101 < Ra < 108.
Theturbulent asymptote of the equations has therefore not beenverified for any fluid.General Correlation for Convex Horizontal Cylinders.For cylinders of arbitrary convex shape, such as shown in Fig.4.19, Eq. 4.48 can be used to find the Nusselt number, proPh p o r t i o n o f Pvided the coefficients G, C2, and C, are known. General~"Pg~ATP 3expressions for these coefficients, based on the recommenNu = ~RaVGtdations of Clemes et al.
[64] and Hassani [125], are given in-P.2ZfFig. 4.19. Note that Nu and Ra in Fig. 4.19 are based on theG = ( 4 Z f / P ) TMC2 = 2nC, = a - (a - b) -~- + e pperimeter of the cylinder, P Z r is the height of the cylinder,a = 0.0972b = 0.0815 - 0.462C vC V is given by Eq. 4.24, and Ph/P is the fraction of theperimeter that lies in the horizontal plane and faces downe = 0 .
6 1 5 C v - 0 . 0 5 4 8 - 6 x 10-6Pr C vFIGURE 4.19 Definition sketch for long horizontal ward for a heated cylinder (it is the fraction that is horizonconvex cylinder of arbitrary shape. The constants G, tal and upward-facing for a cooled cylinder). This correlationgives results in close agreement with Eq. 4.48 (using the conC2, and C, are for use in Eq. 4.48.stants in Table 4.2 and allowing for the different definitions ofNu and Ra). There is also agreement to within about 12 percent with the data of Nakamura andAsako [204] for cylinders of modified square and triangular cross section in the range 3 < Pr _<8./Bodies with Small Aspect RatiosCorrelations for 3D and Axisymmetric Flows. For bodies of "small aspect ratio," there isno single dimension that greatly exceeds all other dimensions.
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