Unformatted text preview: modifying it as
Nu 0.023 Re0.8 Pr n (868) 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 (869) 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 (870) 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 (871)
(872) 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) (873) 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 noslip 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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