0471410772 by Unknown

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§14.6 Heat Transfer Coefficients for Free and Mixed Convection It has been shown, however, that simple and reliable predictions of heat transfer rates (expressed as area mean Nusselt numbers Num) may be obtained for this wide vari- variety of flow regimes by empirical combinations of asymptotic expressions: a. Nu^nd, for conduction in the absence of buoyant forces or forced convection bo Nuj^m, for thin laminar boundary layers, as in Example 11.4-5 c. Nuf"rb, for turbulent boundary layers do Nujj°rced, for pure forced convection These are dealt with in the following subsections. The limiting Nusselt number for vanishingly small free and forced convection is ob- obtained by solving the heat conduction equation (the Laplace equation, V2T = 0) for con- constant, uniform temperature over the solid surface and a different constant temperature at infinity. The mean Nusselt number then has the general form Nucmon = K(shape) A4.6-2) With K equal to zero for all objects with at least one infinite dimension (e.g., infinitely long cylinders or infinitely wide plates). For finite bodies K is nonzero, and an important case is that of the sphere for which, according to Problem 10B.1, ,cond Nu™ = 2 A4.6-3) with the characteristic length taken to be the sphere diameter. Oblate ellipsoids of revo- revolution and circular disks are discussed in Problem 14D.1. For thin laminar boundary layers, the isothermal vertical flat plate is a representative system, conforming to Eq. 14.6-1. This equation may be generalized to = C(Pr, shape)(GrPrI/4 A4.6-4) Moreover, the function of Pr and shape can be factored into the product C - Q(shape)C2(Pr) A4.6-5) with2 0.671 [1 + @.492/Pr)9/16]4/9 Representative values1'3 of Q and C2 are given in Tables 14.6-1 and 2, respectively. Shape factors for a wide variety of other shapes are available.3'4 For heated horizontal flat sur- surfaces facing downward and cooled horizontal flat surfaces facing upward, the following correlation5 is recommended: 2 S. W. Churchill and R. Usagi, AlChE Journal, 23,1121-1128 A972). 3 W. E. Stewart, Int. J. Heat and Mass Transfer, 14,1013-1031 A971). 4 A. Acrivos, AIChE Journal 6, 584-590 A960). 5 T. Fujii, M. Honda, and I. Morioka, Int. J. Heat and Mass Transfer, 15, 755-767 A972).

444 Chapter 14 Interphase Transport in Nonisothermal Systems Table 14.6-1 The Factor Q in Eq. 14.6-5, and the D in the Nusselt Number, for Several Representative Shapes" Shape —> c, "D" in Nu Vertical plate 1.0 Height H Horizontal plate" 0.835 Width W Horizontal cylinder 0.772 Diameter D Sphere 0.878 Diameter D a For a hot upper surface and an insulated lower one, or the reverse for cold surfaces. Table 14,6-2 The Factor C? as a Function of the Prandtl Number Pr C2 Hg 0.022 0.287 Gases 0.71 0.515 1.0 0.534 Water 2.0 4.0 0.568 0.595 6.0 0.608 50 0.650 Oils 100 0.656 2000 0.668 For the vertical plate with a constant-heat-flux boundary condition, the recommended power on GrPr is also 1 /5. Laminar free-convection heat fluxes tend to be small, and a conduction correction is often necessary for accurate predictions.

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