At any point in system through which a fluid is flowing, the total mechanical energy can be expressed in terms of the potential energy, pressure energy, and kinetic energy.

**BERNOULLI’S THEOREM**

At
any point in system through which a fluid is flowing, the total mechanical
energy can be expressed in terms of the potential energy, pressure energy, and
kinetic energy. The potential energy of a body is its capacity to do work by
reason of its position relative to some center of attraction. For unit mass of
fluid at a height z above some reference level, potential energy = *zg*,
where *g* is the acceleration due to
gravity.

The
pressure energy or flow energy is an energy form peculiar to the flow of
fluids. The work done and the energy acquired in transferring the fluid is the
product of the pressure, P, and the volume. The volume of unit mass of the
fluid is the reciprocal of the density, ρ. For an incompressible fluid, the
density is not dependent on the pressure, so for unit mass of fluid, pressure
energy = P/ρ (Fig. 2.5).

The
kinetic energy is a form of energy possessed by a body by reason of its
movement. If the mass of the body is *m*
and its velocity is *u*, the kinetic
energy is 1/2*mu*^{2}, and for
unit mass of fluid, kinetic energy = *u*^{2}/2.

The
total mechanical energy of unit mass of fluid is, therefore,

u^{2}/2 + P/ρ + zg

The
mechanical energy at two points A and B will be the same if no energy is lost
or gained by the system. Therefore, we can write

*FIGURE 2.5** Pressure energy of a
fluid.*

This
relationship neglects the frictional degradation of mechanical energy, which
occurs in real systems. A fraction of the total energy is dissipated in
overcoming the shear stresses induced by velocity gradients in the fluid. If
the energy lost during flow between A and B is E, then equation (2.5) becomes

This
is a form of Bernoulli’s theorem, restricted in application to the flow of
incompressible fluids. Each term is expressed in absolute units, such as
N·m/kg. The dimensions are L^{2}T^{-2}. In practice, each term
is divided by g (LT^{-2}) to give the dimension of length. The terms
are then referred to as velocity head, pres-sure head, potential head, and
friction head, the sum giving the total head of the fluid as shown in equation
(2.2).

The
evaluation of the kinetic energy term requires consideration of the variation
in velocity found in the direction normal to flow. The mean velocity,
calculated by dividing the volumetric flow by the cross-sectional area of the
pipe, lies between 0.5 and 0.82 times the maximum velocity found at the pipe
axis. The value depends on whether flow is laminar or turbulent, terms that are
described later. The mean kinetic energy, given by the term *u*^{2}_{mean}/2 differs
from the true kinetic energy found by summation across the flow direction.

The
former can be retained, however, if a correction factor, *a*, is introduced, then the

velocity
head = *u*^{2}_{mean} / 2g*a*

where
*a* has a value of 0.5 in laminar flow
and approaches unity when flow is fully turbulent.

*FIGURE 2.6** Flow through a constriction.*

A
second modification may be made to equation (2.5) if mechanical energy is added
to the system at some point by means of a pump. If the work done, in absolute
units, on a unit mass of fluid is W, then

The
power required through a system at a certain rate to drive a liquid may be
calculated using equation (2.8). The changes in velocity, pressure, height, and
the mechanical losses due to friction are each expressed as a head of liquid.
The sum of heads, ΔH, being the total
head against which the pump must work, is therefore

W/g + ΔH

If
the work performed and energy acquired by unit mass of fluid is Δ*Hg*, the power required
to transfer mass *m* in time *t* is given by

Power
= ΔH*gm/t*

Since
the volume flowing in unit time Q is m/*ρ**t*,

Power
= QΔ*Hg**ρ*
(2:9)

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