Mechanics of Fluids

254 Flow and losses in pipes and fittings
appreciably affect the value of f in large pipes is waviness of the surface,
that is, the presence of transverse ridges on a larger scale than the normal
roughness.
These and other arguments indicate that Moody’s diagram is at best an
approximation. In addition, the roughness of many materials is very variable
and it frequently increases with age, particularly if the surface becomes dirty
or corroded. (In pipes of small diameter the effective bore may be altered by
dirt and corrosion.) Accurate prediction of friction losses is thus difficult to
achieve.
A number of empirical formulae have been put forward to describe certain
parts of Fig. 7.4. There is, for example, Blasisus’s formula for the turbulent
smooth-pipe curve:
f = 0.079(Re) −1/4
(7.5)
which agrees closely with experimental results for Reynolds number between
4000 and 105 . Many formulae have been proposed so that f can be calculated
directly for the entire range of k/d and Re. The best yet produced is probably
that by S. E. Haaland
�
1
6.9
� =−3.6 log10 f Re +
� � �
1.11
k
3.71d
It combines reasonable simplicity with acceptable accuracy (within about
1.5%) and is useful if an algebraic expression is required.
A fuller discussion of friction factor relationships for flow in smooth or
rough pipes is to be found in Section 8.12.
A common practical problem is to determine the head lost to friction for a
given mean velocity in a pipe of given diameter. From these data the appropriate
Reynolds number may be calculated and a value of f taken from
Fig. 7.4. The head loss is then determined from eqn 7.1. If, however, the
velocity or the diameter is unknown the Reynolds number is also unknown.
Nevertheless, since the value of f changes but slowly with Reynolds number,
assumed values of Re and f may be used for a first trial. Better approximations
to these values can then be obtained from the trial results. This iterative
approach lends itself to solutions using the computer.
For restricted ranges of Reynolds number and pipe diameter, f is sufficiently
constant for tables of values to have been compiled for use in
engineering calculations. Although the use of such values often allows problems
to be solved more simply, limitation of the values to a particular range
of conditions should never be forgotten.
The graph of f as a function of Re and k/d is a convenient presentation
when it is desired to calculate head loss or pressure drop. In the case of inverse
problems, in which the head loss is known and it is required to evaluate some
other variable, such as pipe diameter, flow velocity or volume flow rate
this type of presentation has its drawbacks. The methods of dimensional
analysis provide the answer to these difficulties. Alternative dimensionless
groups can be derived which allow the inverse problems to be solved in a