Mechanics of Fluids

272 Flow and losses in pipes and fittings
as that due to friction in a long uniform pipe) the total head line slopes
uniformly downwards in the direction of flow; where there is a concentrated
dissipation of energy (as at an abrupt change of section) there is an abrupt
step downwards in the total head line. If mechanical energy is added to the
fluid – by means of a pump, for example – the total head line has a step
up. A turbine, on the other hand, which takes energy from the fluid, causes
a step down in the total head line.
In drawing a total head line certain conventional approximations are
made. We have already noted (Section 7.6) that the losses of head arising
from changes of section, bends or other fittings in a pipe are in reality not
concentrated at one point. Nevertheless it is customary to represent the
loss by an abrupt step in the total head line. Also the velocity distribution
across the pipe, which is appropriate to the Reynolds number of the flow
and to the surface roughness, is not achieved immediately the fluid enters the
pipe. The entry length in which the normal pattern of velocity is developed
is often about 50 times the pipe diameter (see Section 7.9) and over this
length the friction factor varies somewhat. For long pipe systems, however,
the entry length is a small proportion of the total length; little error is thus
involved by the usual assumption that the slope of the total head line is
uniform for the entire length of a uniform pipe.
The pressure line (sometimes known as the piezometric line and more
usually in America as the hydraulic grade line) is obtained by plotting values
of p/ϱg vertically above the pipe centre-line. It is therefore a distance u 2 /2g
below the total head line. If the pressure line and the axis of the pipe itself
coincide at any point then the gauge pressure in the pipe is zero, that is,
atmospheric. A pipe that rises above its pressure line has a sub-atmospheric
pressure within it and is known as a siphon. The gauge pressure in any event
cannot fall below −100 kPa (if the atmospheric pressure is 100 kPa) because
that limit would correspond to a perfect vacuum and the flow would stop.
However, the flow of a liquid in a pipe would almost certainly stop before the
pressure fell to this value because dissolved air or other gases would come out
of solution and collect in the highest part of the siphon in sufficient quantity
to form an airlock. Even if the liquid contained no dissolved gases, it would
itself vaporize when the pressure fell to the vapour pressure of the liquid
at that temperature. For these reasons the pressure of water in a pipe-line
should not fall below about −75 kPa gauge. This suggests that the pressure
line should not be more than about
75 000 N · m−2 1000 kg · m−3 = 7.64 m
−1
9.81 N · kg
below the pipe, but siphons with somewhat lower pressure lines are possible
because the emerging gas bubbles reduce the mean density of the
liquid.
The air in a siphon must of course be extracted by some means in order to
start the flow of liquid, and air subsequently collecting at the highest point
must be removed if flow is to continue. Automatic float valves may be used