Page 91 - Mechanical Engineers' Handbook (Volume 4)

80 Fluid Mechanics

keeping the ratio of the two constant. A relation between the centerline velocity and the
average velocity is u max /V 1 133 ƒ, which may be used to estimate the average velocity
from a single centerline measurement.
The Colebrook–White equation encompasses all turbulent flow regimes, for both smooth
and rough pipes:
1.74 2 log
1 2k 18.7
ƒ D Re ƒ
D
and this is plotted in Fig. 32, where k is the equivalent sand-grain roughness. A simpler
equation by Haaland is
1.8 log
1.11
1 6.9 k
ƒ Re D 3.7D
which is explicit in ƒ and is within 1.5% of the Colebrook–White equation in the range 4000
8
Re 10 and 0 k/D 0.05.
D
Three types of problems may be solved:
1. The pressure drop or head loss. The Reynolds number and relative roughness are
determined and calculations are made directly.
2. The flow rate for given fluid and pressure drops or head loss. Assume a friction
factor, based on a high Re for a rough pipe, and determine the velocity from the
D
Darcy equation. Calculate a Re , get a better ƒ, and repeat until successive velocities
D
are the same. A second method is to assume a flow rate and calculate the pressure
drop or head loss. Repeat until results agree with the given pressure drop or head
loss. A plot of Q versus h , for example, for a few trials may be used.
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3. A pipe size. Assume a pipe size and calculate the pressure drop or head loss. Com-
pare with given values: Repeat until agreement is reached. A plot of D versus h ,
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for example, for a few trials may be used. A second method is to assume a reasonable
friction factor and get a first estimate of the diameter from
D 1/5
2
8ƒLQ
2
gh ƒ
From the first estimate of D, calculate the Re and k/D to get a better value of ƒ.
D
Repeat until successive values of D agree. This is a rapid method.
Results for circular pipes may be applied to noncircular ducts if the hydraulic diameter
is used in place of the diameter of a circular pipe. Then the relative roughness is k/D and
h
the Reynolds number is Re VD /v. Results are reasonably good for square ducts, rectan-
h
gular ducts of aspect ratio up to about 8, equilateral ducts, hexagonal ducts, and concentric
annular ducts of diameter ratio to about 0.75. In eccentric annular ducts where the pipes
touch or nearly touch, and in tall narrow triangular ducts, both laminar and turbulent flow
may exist at a section. Analyses mentioned here do not apply to these geometries.

11.4 Steady Incompressible Flow in Entrances of Ducts
The increased pressure drop in the entrance region of ducts as compared with that for the
same length of fully developed flow is generally included in a correction term called a loss
coefficient, k . Then,
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