Mechanics of Fluids

General energy equation for steady flow of any fluid 103
With this particular distribution of velocity over the cross-section the term Further discussion of
u Example 3.1
2 /2 is therefore about 6% too low to represent the mean kinetic energy
divided by mass. It is usual, however, to disregard such discrepancy except
where great accuracy is required. In any case, the exact value of the correction
to be applied is much influenced by conditions upstream and is scarcely ever
known. The correction should nevertheless be remembered if the Steady-
Flow Energy Equation is applied to fully developed laminar flow in a circular
pipe (see Chapter 6), for then the mean kinetic energy divided by mass =
2u2 /2. Even so, as laminar flow is generally associated only with very low
velocities the kinetic energy term would in these circumstances probably be
negligible.
3.5.4 The Steady-Flow Energy Equation in practice
The Steady-Flow Energy Equation (SFEE) applies to liquids, gases and
vapours, and accounts for viscous effects. In many applications it is considerably
simplified because some of the terms are zero or cancel with others.
If no heat energy is supplied to the fluid from outside the boundaries, and
if the temperature of the fluid and that of its surroundings are practically
identical (or if the boundaries are well insulated) q may be taken as zero.
If there is no machine between sections (1) and (2) the shaft work divided
by mass w is zero. And for fluids of constant density ϱ1 = ϱ2.
If an incompressible fluid with zero viscosity flows in a stream-tube across
which there is no transfer of heat or work, the temperature of the fluid
remains constant. Therefore the internal energy is also constant and the
equation reduces to
0 =
� p2
ϱ2
+ 1
2 u2 � �
p1
2 + gz2 − +
ϱ1
1
2 u2 �
1 + gz1
This is seen to be identical with Bernoulli’s equation (3.9).
Real fluids have viscosity, and the work done in overcoming the viscous
forces corresponds to the so-called fluid friction. The energy required to
overcome the friction is transformed into thermal energy. The temperature
of the fluid rises above the value for frictionless flow; the internal energy
increases and, in general, the heat transferred from the fluid to its surroundings
is increased. The increase of temperature, and consequently of internal
energy, is generally of no worth (the temperature rise is normally only a very
small fraction of a degree) and thus corresponds to a loss of useful energy.
Moreover, as we have defined q as the heat transferred to the fluid divided
by the mass of the fluid, a loss of heat from the system is represented by −q
and so the total loss (divided by the mass of the fluid) is e2 − e1 − q. For
a fluid of constant density it is usual to express this loss of useful energy,
resulting from friction, as head loss due to friction, h f. Therefore
h f = (e2 − e1 − q)/g