Bernoulli's Equation
Energy Equation of an ideal Flow along a Streamline
Euler’s equation (the equation of motion of an inviscid fluid) along a stream line for a steady flow with gravity as the only body force can be written as

(13.6) 
Application of a force through
a distance ds along the streamline would physically imply
work interaction. Therefore an equation for conservation of
energy along a streamline can be obtained by integrating the
Eq. (13.6) with respect to ds as



(13.7) 
Where C is a constant along
a streamline. In case of an incompressible flow, Eq. (13.7)
can be written as

(13.8) 
The Eqs (13.7) and (13.8) are
based on the assumption that no work or heat interaction between
a fluid element and the surrounding takes place. The first
term of the Eq. (13.8) represents the flow work per unit mass,
the second term represents the kinetic energy per unit mass
and the third term represents the potential energy per unit
mass. Therefore the sum of three terms in the left hand side
of Eq. (13.8) can be considered as the total mechanical energy
per unit mass which remains constant along a streamline for
a steady inviscid and incompressible flow of fluid. Hence
the Eq. (13.8) is also known as Mechanical
energy equation.
This equation was developed first by Daniel Bernoulli in 1738
and is therefore referred to as Bernoulli’s equation.
Each term in the Eq. (13.8) has the dimension of energy per
unit mass. The equation can also be expressed in terms of
energy per unit weight as

(13.9) 
In a fluid flow, the energy
per unit weight is termed as head. Accordingly, equation 13.9
can be interpreted as
Pressure head + Velocity head + Potential head =Total head (total energy per unit weight).
Bernoulli's Equation with Head Loss
The derivation of mechanical energy equation for a real fluid depends much on the information about the frictional work done by a moving fluid element and is excluded from the scope of the book. However, in many practical situations, problems related to real fluids can be analysed with the help of a modified form of Bernoulli’s equation as

(13.10) 
where, h_{f}
represents the frictional work done (the work done against
the fluid friction) per unit weight of a fluid element while
moving from a station 1 to 2 along a streamline in the direction
of flow. The term h_{f} is usually referred to as
head loss between 1 and 2, since it amounts to the loss in
total mechanical energy per unit weight between points 1 and
2 on a streamline due to the effect of fluid friction or viscosity.
It physically signifies that the difference in the total mechanical
energy between stations 1 and 2 is dissipated into intermolecular
or thermal energy and is expressed as loss of head h_{f}
in Eq. (13.10). The term head loss, is conventionally symbolized
as h_{L} instead of h_{f} in dealing with
practical problems. For an inviscid flow h_{L} = 0,
and the total mechanical energy is constant along a streamline.
End of Lecture 13!
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