Fluid Flow
BERNOULLI’S EQUATION
Multiplying all terms in Equation 3-10 by the factor gc/mg results in the form of Bernoulli’s
equation shown by Equation 3-11.
(3-11)
z1
v2
1
2g
P1n1
gc
g
z2
v2
2
2g
P2n2
gc
g
Head
Since the units for all the different forms of energy in Equation 3-11 are measured in units of
distance, these terms are sometimes referred to as "heads" (pressure head, velocity head, and
elevation head). The term head is used by engineers in reference to pressure. It is a reference
to the height, typically in feet, of a column of water that a given pressure will support. Each of
the energies possessed by a fluid can be expressed in terms of head. The elevation head
represents the potential energy of a fluid due to its elevation above a reference level. The
velocity head represents the kinetic energy of the fluid. It is the height in feet that a flowing
fluid would rise in a column if all of its kinetic energy were converted to potential energy. The
pressure head represents the flow energy of a column of fluid whose weight is equivalent to the
pressure of the fluid.
The sum of the elevation head, velocity head, and pressure head of a fluid is called the total
head. Thus, Bernoulli’s equation states that the total head of the fluid is constant.
Energy Conversions in Fluid Systems
Bernoulli’s equation makes it easy to examine how energy transfers take place among elevation
head, velocity head, and pressure head. It is possible to examine individual components of piping
systems and determine what fluid properties are varying and how the energy balance is affected.
If a pipe containing an ideal fluid undergoes a gradual expansion in diameter, the continuity
equation tells us that as the diameter and flow area get bigger, the flow velocity must decrease
to maintain the same mass flow rate. Since the outlet velocity is less than the inlet velocity, the
velocity head of the flow must decrease from the inlet to the outlet. If the pipe lies horizontal,
there is no change in elevation head; therefore, the decrease in velocity head must be
compensated for by an increase in pressure head. Since we are considering an ideal fluid that
is incompressible, the specific volume of the fluid will not change. The only way that the
pressure head for an incompressible fluid can increase is for the pressure to increase. So the
Bernoulli equation indicates that a decrease in flow velocity in a horizontal pipe will result in an
increase in pressure.
If a constant diameter pipe containing an ideal fluid undergoes a decrease in elevation, the same
net effect results, but for different reasons. In this case the flow velocity and the velocity head
must be constant to satisfy the mass continuity equation.
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