Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 25
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The ratio of the values for the sphere a/b = 1and the long prolate spheroid a/b = 10 has been reduced from 3.0 to 1.18.The numerical values for elliptical disks are presented in the third column of Table 3.3.Here, also, we observe that the reduction in the range is much greater. The ratio of the valuesfor the circular disk a/b = 1 and the long elliptical disk a/b = 10 has been reduced from 2.35to 1.34.There is another benefit when ~ = X/A is used as the characteristic body length. The differences between the values for the elliptical disks and the prolate spheroids are greatly reduced,becoming negligible for large aspect ratios.
The largest difference of approximately 11 percentoccurs when the aspect ratio is 1, i.e., when a sphere and a circular disk are compared.This shows that elliptical disks (zero-thickness bodies) and prolate spheroids that haveidentical total surface areas and similar aspect ratios possess shape factors that are close inmagnitude.This important finding is demonstrated further in the subsequent sections, where a widerange of body shapes is considered.Raithby and Hollands, in the chapter on natural convection, have developed correlationequations for external convection from isothermal bodies. In the correlation equations, theconduction Nusselt number is based on the shape factors developed in these sections.Shape Factors for Three-Dimensional Bodies in Unbounded DomainsThe shape factors for steady conduction within two- and three-dimensional systems that arebounded by isothermal surfaces are available.
Dimensionless shape factors for several threedimensional bodies are presented next. The results are based on analytical and numericaltechniques.Circular Toroid. The circular toroid is characterized by the ring diameter d and the toroiddiameter D (Fig. 3.4a). The analytical solution [94] for the shape factor is written as an infiniteseries in which each term consists of toroidal or ring functions [1]:S ~ A - ~Q/ ' A- 4-N1 It,/ ~/ - 2I / ~({ ~ ) P-1/2(~)~+2n=~Qn-1/2(~)}enn--~(3.42)with ~ - D/d > 1. The special functions are accurately computed using Mathematica. The seriesconverges very slowly for D/d ~ 1, which corresponds to toroids with small inner diameters.When D/d = 1, S~A = 3.482761, which is approximately 1.8 percent smaller than the value forthe sphere.
In the narrow range: 1 < D/d < 2, S~/~-A= 3.449 to within 1 percent.Doi°--'11--- o----1Do(a)(b)0 I--_ wk_(c)FIGURE 3.4 (a) Circular toroid; (b) square toroid; (c) finite circular cylinder; (d) finite square cylinder.(d)3.16CHAPTERTHREESINGLECONEiHEMISPHEREI_DI-_]-IDOUBLECONE_BISPHERE(e)D_I(f)FIGURE 3.4 (Continued) (e) bisphere and hemisphere; (f) single anddouble cones.For D/d > 5, the dimensionless shape factor for the toroid approaches the asymptote:v~x/b-7~S~-x- - - - 2 nAIn (8D/d)(3.43)For the practical range 2 g D/d < 10, the shape factor is approximated with a maximumerror of about 0.7 percent by multiplying the asymptotic expression by the empirical correlation equation:81e -D/dCc~=~+Square Toroid.
The accurate numerical values of shape factors for square toroids, which arecharacterized by the inner and outer diameters Di and Do, respectively (Fig. 3.4b), werereported by Wang [122] for a wide range of the diameter ratio Di/Do. The dimensionlessresults were found to be in close agreement with the analytical results for the equivalent circular toroid defined byDx 1 + Di/Dod - 4 1 - D,/Do'D~Do > 0.1(3.44)The above relationship was obtained by the application of two geometric rules: (1) set the surface area and (2) set the mean perimeter of the equivalent circular toroid equal to the surfacearea and the mean perimeter of the square toroid.
When the circular toroid asymptoticexpression given above is multiplied by the empirical correlation equationCONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)1613.17e -o/dcs~=Tff6 + ~with D/d defined by the previous equation, the numerical values and the predicted values differ by less than 0.8 percent.The dimensionless shape factor for an isothermal right circularcylinder of length L and diameter d (Fig. 3.4c) was obtained from the analytical solution forthe capacitance [96, 97].
Using the square root of the total surface area, the result is recast asFinite Circular Cylinder.3.1915 + 2.7726(L/d) °76S~A -A-~/1 + 2L/d,0 < L/d < 8(3.45)The dimensionless shape factor for the right circular cylinder is in very close agreement withthe values for the oblate spheroid in the range 0 < L/d < 1 and with the values for the prolatespheroid in the range 1 < L/d _<8.
The difference when L/d = 1 is less than 1 percent. Thisshows that the results for the sphere and a finite circular cylinder of unit aspect ratio are veryclose. The simple expression obtained from the Smythe solution can be used to estimate theshape factors of circular disks, oblate spheroids, and prolate spheroids in the range 0 , L/d <8. For L/d > 8, the prolate spheroid asymptotic result can be used to provide accurate resultsfor long circular cylinders and other equivalent bodies.The dimensionless shape factors for finite square cylinders oflength L and side dimension W (Fig. 3.4d) can be calculated using the finite circular expression by means of the equivalent aspect ratioFinite Square Cylinder.Ld1L2 TM W(3.46)which was obtained by the following rule: take the geometric mean of the aspect ratiosobtained by (1) inscribing and (2) circumscribing the square cylinder with circular cylinders.This procedure produces results that are in close agreement with reported numerical resultsin the range 0 < L / W < 10.Cube, Bisphere, a n d Hemisphere.
The dimensionless shape factors are as follows: for thecube (Fig. 3.3), S~A = 3.391, for the bisphere (Fig. 3.4e), S~A = 3.4749, and for the hemisphere(Fig. 3.4e), S~A = 3.4601. These numerical and analytical results are approximately 4.5 percent, 2 percent, and 2.5 percent smaller than the value for the sphere.The single and double cones (Fig. 3.4f) are characterized by theheight dimension H and the maximum diameter D. The dimensionless shape factors wereobtained by means of an accurate numerical technique [122].Single cone. The single cone numerical results for the shape factor are predicted by thefollowing two correlation equations.Single a n d D o u b l e Cones.for 0.001< x =H / D < 1:Sv~ = 3.19399 + 0.629823x - 0.933731x 2 + 0.862597x 3 - 0.312459x 4for1 _< x =(3.47)H/D <_8:S~-A = 3.280967 + 1.61022(X/10) -0.047366(X/10) 2 - 0.30067(x/10) 3 + 2.99117 x 10-3(x/10) 4(3.48)3.18CHAPTERTHREEDouble Cone.
The double cone numerical data for the shape factors are correlated bythe following two polynomials.for 0.001 < x = H/D < 1:S ~ = 3.194264 + 0.626604x - 0.477791x 2 + 0.0751056x 3 + 0.0531827x 4(3.49)and for 1 < x = H/D <_10:S~m = 3.41318 + 0.419048(X/10) + 2.02734(X/10) 2 -- 2.23961(X/10) 3 + 0.80661(X/10) 4 (3.50)Circular and Rectangular Annulus. The dimensionless shape factors of isothermal circularand rectangular annuli are presented next.Circular annulus. The circular annulus has inner and outer radii a and b, respectively. Thetwo capacitance analytical solutions of Smythe [95] are recast into the following two expressions, which relate the dimensionless shape factor to the radii ratio e = a/b:S~a _ V/-A- _ r r V ~ 7 iA+_e .1- e In 16 + In [(1 + e)/(1 - e)](3.51)which is restricted to the range 1.000 < 1/e < 1.1; andr - - - - -S~-xa_ --_VA 8 7 2A~1V I -~[cos -~ e + V'I - e2 tanh -~ e][1 + 0.0143e -~ tan 3 (1.28e)](3.52)which is valid in the range 0 < e < 1/1.1.Rectangular annulus.
The rectangular annulus is characterized by its outer length L andouter width W. The width W of the annular area is uniform. The interior open region hasdimensions L - 2°14/`by W - 2°H/".The dimensionless shape factor for the isothermal rectangular annulus is derived from thecorrelation equation of Schneider [89], who obtained accurate numerical values of the thermal constriction resistance of doubly connected rectangular contact areas by means of theboundary integral equation method:S{Fx~/AA~[WL- [CI(W/2W)C2+ C3 ] " [-W + W -/ W \21-1'22[-~) J(3.53)with the recommended correlation coefficientsC 1=L-0.00232 --~ + 0.03128C2 = 0.2927+ 0.7463C3 = 0.6786+ 0.8145)')'+ 0.4316+ 0.0346The correlation equation is restricted to the ranges 1 _<L / W < 4 and 0.01 < ~ / W < 0.5, witha maximum error of 1.45 percent between the correlation predictions and the numerical values.
Selected values of the dimensionless diffusion length ~/-A/A are given in Table 3.4.The dimensionless diffusion length is a weak function of the two parameters L / W and74r/w over a practical range of these parameters. The values are close to the circular and elliptical disk values reported earlier.CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)TABLE 3.43.19Dimensionless Diffusion Length for Rectangular AnnulusL/WW/WV/--A/AL/WW/W~/-A/A10.10.30.53.29183.19783.357530.10.30.53.70013.61063.637020.10.30.53.48933.40443.468140.10.30.53.87743.78293.7884Rectangular Plate.The dimensionless shape factor for isothermal rectangular plates oflength L and width W, where L >_W, is calculated by means of the following two semianalytical expressions:(1 + V'L/W) 2S~A = 0.8V'L/W'1 < L/W < 5(3.54)k/8nL/WS~AA= In (4L/W) '5 < L/W < oo(3.55)The maximum difference between the predicted values and the numerical values for therange 1 _<L/W_< 4 is less than 1 percent.Cuboid.
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