The Bernoulli’s equation is one of the most useful equations
that is applied in a wide variety of fluid flow related
problems. This equation can be derived in different
ways, e.g. by integrating Euler’s equation along
a streamline, by applying first and second laws of
thermodynamics to steady, irrotational, inviscid
and in-compressible flows etc. In simple form
the Bernoulli’s equation relates the pressure, velocity and
elevation between any two points in the flow
field. It is a scalar equation and is
given by:
Basic Bernoulli’s Equation |
Each term in the above equation has dimensions of length
(i.e., meters in SI units) hence these terms are called as
pressure head, velocity head, static head
and total heads respectively.
Bernoulli’s equation can also be written in terms of
pressures (i.e.,Pascals in SI units) as:
Bernoulli’s equation in terms Of Pressure |
Bernoulli’s equation is valid between any two
points in the flow field when the flow is
steady, irrotational, in-viscid and incompressible.
The equation is valid along a streamline for rotational,
steady and incompressible flows. Between any two points
1 and 2 in the flow field for irrotational flows, the
Bernoulli’s equation is written as:
Bernoulli’s equation with Datum |
Bernoulli’s equation can also be considered to be an
alternate statement of conservation of energy
(1st
law of thermodynamics).
The equation also implies the possibility of conversion of
one form of pressure into other. For example,
neglecting the pressure changes due to datum, it can be
concluded from Bernoulli’s equation that the static
pressure rises in the direction of flow in a diffuser while it
drops in the direction of flow in case of nozzle due to
conversion of velocity pressure into static pressure and
vice versa. Figure 1 shows the
variation of total, static and velocity pressure for
steady, incompressible and inviscid, fluid flow through a pipe
of uniform cross-section.
Since all real fluids have finite viscosity, i.e. in all
actual fluid flows, some energy will be lost in
overcoming friction. This is referred to as head loss, i.e.
if the fluid were to rise in a vertical pipe it will
rise to a lower height than predicted by Bernoulli’s
equation. The head loss will cause the pressure to decrease
in the flow
direction. If the head loss is denoted by H then
Bernoulli’s equation can be modified to:
Fig 1. Application Of Bernoulli Equation. |
Figure 1 shows the variation of total, static and velocity
pressure for steady, incompressible fluid flow through a
pipe of uniform cross-section without viscous effects
(solid line) and with viscous effects (dashed lines).