Mechanics of Fluids

248 Flow and losses in pipes and fittings
Friction factor
Such a low velocity is seldom of interest in practice. A critical velocity 10
times as great would require a diameter 10 times smaller, that is, 2.5 mm.
This, too, is a figure outside normal engineering practice. The velocity of
water is generally far greater than the critical value and so the flow is normally
fully turbulent. This is not to say that laminar flow of water may not
be found in laboratory experiments or other small-scale work. Indeed, in the
testing of small-scale models of hydraulic structures, the presence of laminar
flow that does not correspond to the flow occurring in the full-size prototype
may constitute a considerable difficulty.
Water, however, has a low viscosity. Oil having a kinematic viscosity of,
say, 200 times that of water would, in a pipe of 25 mm diameter, have a
critical velocity of 200 × 0.092 m · s −1 = 18.4 m · s −1 . This velocity is far in
excess of any that one would expect in practice, so it would be necessary to
treat the flow of such oil as laminar. Even in a pipe of 250 mm diameter, a
mean velocity of 1.84 m · s −1 could be reached without the flow becoming
turbulent.
By similar considerations we may see that the wholly laminar flow of air
or steam in a pipe does not often occur.
7.2.2 Head lost to friction
One of the most important items of information the engineer needs is the
pressure difference, or difference of piezometric head, required to induce
fluid to flow at a certain steady rate through a pipe. About the middle of
the nineteenth century, therefore, many experimenters devoted attention to
this topic. Among them was the French engineer Henri Darcy (1803–58)
who investigated the flow of water, under turbulent conditions, in long,
unobstructed, straight pipes of uniform diameter. The fall of piezometric
head in the direction of flow is caused by the dissipation of energy by
fluid friction. If the pipe is of uniform cross-section and roughness, and
the flow is fully developed, that is if it is sufficiently far from the inlet
of the pipe for conditions to have become settled, the piezometric head
falls uniformly. Darcy’s results suggest the formula (now commonly named
after him):
h f = �p∗
ϱg
= 4fl
d
u 2
2g
(7.1)
In eqn 7.1, h f represents the head lost to friction, corresponding (in steady
flow) to the drop �p ∗ of piezometric pressure over length l of the pipe, ϱ
represents the density of the fluid, u the mean velocity (i.e. discharge divided
by cross-sectional area), f is a coefficient, g the gravitational acceleration
and d the pipe diameter.
The coefficient f in eqn 7.1 is usually known as the friction factor. Comparison
of the dimensional formulae of the two sides of eqn 7.1 shows that f is
simply a numeric without units. Observations show that its value depends on
the Reynolds number of the flow and on the roughness of the pipe surface. In