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Unformatted text preview: e Nu relations above should be evaluated at the bulk mean fluid temperature, which is the arithmetic average of the
mean fluid temperatures at the inlet and the exit of the tube. For laminar flow,
the effect of surface roughness on the friction factor and the heat transfer coefficient is negligible. Laminar Flow in Noncircular Tubes
The friction factor f and the Nusselt number relations are given in Table 8–1
for fully developed laminar flow in tubes of various cross sections. The
Reynolds and Nusselt numbers for flow in these tubes are based on the hydraulic diameter Dh 4Ac /p, where Ac is the cross sectional area of the tube
and p is its perimeter. Once the Nusselt number is available, the convection
heat transfer coefficient is determined from h kNu/Dh. Developing Laminar Flow in the Entrance Region
For a circular tube of length L subjected to constant surface temperature, the
average Nusselt number for the thermal entrance region can be determined
from (Edwards et al., 1979)
Entry region, laminar: Nu 3.66 1 0.065 (D/L) Re Pr
0.04[(D/L) Re Pr]2/3 (862) cen58933_ch08.qxd 9/4/2002 11:29 AM Page 437 437
CHAPTER 8 TABLE 8–1
Nusselt number and friction factor for fully developed laminar flow in tubes of
various cross sections (Dh 4Ac /p, Re
hDh /k)
m Dh /v, and Nu Tube Geometry
Circle a/b
or ° Ts — Nusselt Number
·
Const.
q s Const. Friction Factor
f 3.66 4.36 64.00/Re 2.98
3.39
3.96
4.44
5.14
5.60
7.54 3.61
4.12
4.79
5.33
6.05
6.49
8.24 56.92/Re
62.20/Re
68.36/Re
72.92/Re
78.80/Re
82.32/Re
96.00/Re a/b
1
2
4
8
16 3.66
3.74
3.79
3.72
3.65 4.36
4.56
4.88
5.09
5.18 64.00/Re
67.28/Re
72.96/Re
76.60/Re
78.16/Re 10°
30°
60°
90°
120° 1.61
2.26
2.47
2.34
2.00 2.45
2.91
3.11
2.98
2.68 50.80/Re
52.28/Re
53.32/Re
52.60/Re
50.96/Re D Rectangle b
a Ellipse b
a a/b
1
2
3
4
6
8 Triangle θ Note that the average Nusselt number is larger at the entrance region, as expected, and it approaches asymptotically to the fully developed value of 3.66
as L → . This relation assumes that the flow is hydrodynamically developed
when the fluid enters the heating section, but it can also be used approximately for flow developing hydrodynamically.
When the difference between the surface and the fluid temperatures is large,
it may be necessary to account for the variation of viscosity with temperature.
The average Nusselt number for developing laminar flow in a circular tube in
that case can be determined from [Sieder and Tate (1936), Ref. 26]
Nu 1.86 Re Pr D
L 1/3 b
s 0.14 (863) All properties are evaluated at the bulk mean fluid temperature, except for
which is evaluated at the surface temperature. s, cen58933_ch08.qxd 9/4/2002 11:29 AM Page 438 438
HEAT TRANSFER The average Nusselt number for the thermal entrance region of flow
between isothermal parallel plates of length L is expressed as (Edwards
et al., 1979)
Entry region, laminar: Nu 7.54 1 0.03 (Dh /L) Re Pr
0.016[(Dh /L) Re Pr]2/3 (864) where Dh is the hydraulic diameter, which is twice the spacing of the plates.
This relation can be used for Re 2800.
EXAMPLE 8–2 3 ft/s 0.15 in. Pressure Drop in a Pipe Water at 40°F (
62.42 lbm/ft3 and
3.74 lbm/ft h) is flowing in a 0.15in.diameter 30ftlong pipe steadily at an average velocity of 3 ft/s (Fig. 8–22).
Determine the pressure drop and the pumping power requirement to overcome
this pressure drop. 30 ft FIGURE 8–22
Schematic for Example 8–2. SOLUTION The average flow velocity in a pipe is given. The pressure drop and
the required pumping power are to be determined.
Assumptions 1 The flow is steady and incompressible. 2 The entrance effects
are negligible, and thus the flow is fully developed. 3 The pipe involves no components such as bends, valves, and connectors.
Properties The density and dynamic viscosity of water are given to be
62.42 lbm/ft3 and
3.74 lbm/ft h 0.00104 lbm/ft s.
Analysis First we need to determine the flow regime. The Reynolds number is
mD Re (62.42 lbm/ft3)(3 ft/s)(0.12/12 ft) 3600 s
3.74 lbm/ft · h
1h 1803 which is less than 2300. Therefore, the flow is laminar. Then the friction factor
and the pressure drop become f 64
Re 64
1803 0.0355 2 P 30 ft (62.42 lbm/ft3)(3 ft/s)2
1 lbf
m
L
0.0355
D2
2
0.12/12 ft
32.174 lbm · ft/s2
930 lbf/ft2 6.46 psi
f The volume flow rate and the pumping power requirements are
·
2
V
(3 ft/s)[ (0.12/12 ft)2/4] 0.000236 ft3/s
m Ac
m ( D /4)
·
Wpump ·
VP (0.000236 ft3/s)(930 lbf/ft2) 1W
0.737 lbf · ft/s 0.30 W Therefore, mechanical power input in the amount of 0.30 W is needed to overcome the frictional losses in the flow due to viscosity. cen58933_ch08.qxd 9/4/2002 11:29 AM Page 439 439
CHAPTER 8 EXAMPLE 8–3 Flow of Oil in a Pipeline through a Lake Consider the flow of oil at 20°C in a 30cmdiameter pipeline at an average
velocity of 2 m/s (Fig. 8–23). A 200mlong section of the pipeline passes
through icy waters of a lake at 0°C. Measurements indicate that the surface
temperature of the pipe is very nearly 0°C. Disregarding the thermal resistance
of the pipe mat...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.
 Spring '10
 Ghaz

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