866 it gives nu 07 re 0023 re08 pr13 pr 160 10000 8

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Unformatted text preview: modifying it as Nu 0.023 Re0.8 Pr n (8-68) where n 0.4 for heating and 0.3 for cooling of the fluid flowing through the tube. This equation is known as the Dittus–Boelter equation [Dittus and Boelter (1930), Ref. 6] and it is preferred to the Colburn equation. The fluid properties are evaluated at the bulk mean fluid temperature Tb (Ti Te)/2. When the temperature difference between the fluid and the wall is very large, it may be necessary to use a correction factor to account for the different viscosities near the wall and at the tube center. The Nusselt number relations above are fairly simple, but they may give errors as large as 25 percent. This error can be reduced considerably to less than 10 percent by using more complex but accurate relations such as the second Petukhov equation expressed as Nu 1.07 ( f /8) Re Pr 12.7( f /8)0.5 (Pr2/3 1) 0.5 104 Pr Re 2000 5 106 (8-69) The accuracy of this relation at lower Reynolds numbers is improved by modifying it as [Gnielinski (1976), Ref. 8] Nu ( f /8)(Re 1000) Pr 1 12.7( f /8)0.5 (Pr2/3 1) 0.5 Pr 3 103 2000 Re 5 106 (8-70) cen58933_ch08.qxd 9/4/2002 11:29 AM Page 442 442 HEAT TRANSFER Relative Roughness, /L Friction Factor, f 0.0* 0.00001 0.0001 0.0005 0.001 0.005 0.01 0.05 where the friction factor f can be determined from an appropriate relation such as the first Petukhov equation. Gnielinski’s equation should be preferred in calculations. Again properties should be evaluated at the bulk mean fluid temperature. The relations above are not very sensitive to the thermal conditions at the · tube surfaces and can be used for both Ts constant and qs constant cases. Despite their simplicity, the correlations already presented give sufficiently accurate results for most engineering purposes. They can also be used to obtain rough estimates of the friction factor and the heat transfer coefficients in the transition region 2300 Re 10,000, especially when the Reynolds number is closer to 10,000 than it is to 2300. The relations given so far do not apply to liquid metals because of their very low Prandtl numbers. For liquid metals (0.004 Pr 0.01), the following relations are recommended by Sleicher and Rouse (1975, Ref. 27) for 104 Re 106: 0.0119 0.0119 0.0134 0.0172 0.0199 0.0305 0.0380 0.0716 *Smooth surface. All values are for Re and are calculated from Eq. 8–73. FIGURE 8–24 The friction factor is minimum for a smooth pipe and increases with roughness. Standard sizes for Schedule 40 steel pipes Nominal Size, in. Actual Inside Diameter, in. ⁄8 ⁄4 3 ⁄8 1 ⁄2 3 ⁄4 1 11⁄2 2 21⁄2 3 5 10 0.269 0.364 0.493 0.622 0.824 1.049 1.610 2.067 2.469 3.068 5.047 10.02 1 constant: constant: Nu Nu 4.8 6.3 0.0156 Re0.85 Pr0.93 s 0.0167 Re0.85 Pr0.93 s (8-71) (8-72) where the subscript s indicates that the Prandtl number is to be evaluated at the surface temperature. Rough Surfaces 6 10 , Any irregularity or roughness on the surface disturbs the laminar sublayer, and affects the flow. Therefore, unlike laminar flow, the friction factor and the convection coefficient in turbulent flow are strong functions of surface roughness. The friction factor in fully developed turbulent flow depends on the Reynolds number and the relative roughness /D. In 1939, C. F. Colebrook (Ref. 3) combined all the friction factor data for transition and turbulent flow in smooth as well as rough pipes into the following implicit relation known as the Colebrook equation. 1 TABLE 8–2 1 Liquid metals, Ts Liquid metals, q·s f 2.0 log /D 3.7 2.51 Re f (turbulent flow) (8-73) In 1944, L. F. Moody (Ref. 17) plotted this formula into the famous Moody chart given in the Appendix. It presents the friction factors for pipe flow as a function of the Reynolds number and /D over a wide range. For smooth tubes, the agreement between the Petukhov and Colebrook equations is very good. The friction factor is minimum for a smooth pipe (but still not zero because of the no-slip condition), and increases with roughness (Fig. 8–24). Although the Moody chart is developed for circular pipes, it can also be used for noncircular pipes by replacing the diameter by the hydraulic diameter. At very large Reynolds numbers (to the right of the dashed line on the chart) the friction factor curves corresponding to specified relative roughness curves are nearly horizontal, and thus the friction factors are independent of the Reynolds number. In calculations, we should make sure that we use the internal diameter of the pipe, which may be different than the nominal diameter. For example, the internal diameter of a steel pipe whose nominal diameter is 1 in. is 1.049 in. (Table 8–2). cen58933_ch08.qxd 9/4/2002 11:29 AM Page 443 443 CHAPTER 8 Commercially available pipes differ from those used in the experiments in that the roughness of pipes in the market is not uniform, and it is difficult to give a precise description of it. Equivalent roughness values for some commercial pipes are given in Table 8–3, as well as on the Moody chart. But it should be kept in mind that these values are for n...
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