Mechanics of Fluids

262 Flow and losses in pipes and fittings
Exit loss
Fig. 7.9
From the energy equation for a constant density fluid we have
p1
ϱg + u2 1
2g + z − hl = p2
ϱg + u2 2
+ z
2g
where h l represents the loss of total head between sections 1 and 2.
and substitution from eqn 7.13 gives
∴ hl = p1 − p2
ϱg + u2 1 − u2 2
2g
h l = u2(u2 − u1)
g
+ u2 1 − u2 2
2g
= (u1 − u2) 2
2g
Since by continuity A1u1 = A2u2 eqn 7.14 may be alternatively written
hl = u2 �
1
1 −
2g
A1
A2
� 2
= u2 �
2 A2
2g A1
− 1
� 2
(7.14)
(7.15)
This result was first obtained by J.-C. Borda (1733–99) and
L. M. N. Carnot (1753–1823) and is sometimes known as the Borda–Carnot
head loss. In view of the assumptions made, eqns 7.14 and 7.15 are subject
to some inaccuracy, but experiments show that for coaxial pipes they are
within only a few per cent off the truth.
If A2 →∞, then eqn 7.15 shows that the head loss at an abrupt enlargement
tends to u2 1 /2g. This happens at the submerged outlet of a pipe discharging
into a large reservoir, for example (Fig. 7.9), or for a duct discharging
to atmosphere. The velocity head in the pipe, corresponding to the kinetic
energy of the fluid divided by weight, is thus lost in turbulence in the
reservoir. In such circumstances the loss is usually termed the exit loss for
the pipe.
7.6.2 Loss at abrupt contraction
Although an abrupt contraction (Fig. 7.10) is geometrically the reverse of an
abrupt enlargement it is not possible to apply the momentum equation to a
control volume between sections 1 and 2. This is because, just upstream of