

In this section, Bernoulli's equation will
be introduced. The details of the derivation are simplified, with
attention focused on proper use of the equation. Restrictions
on the
application of Bernoulli's equation are also clearly stated to avoid
misuse of
the equation. A velocity measurement device called a Pitot tube will
also be presented. In addition, the concept of energy and hydraulic grade
lines will be introduced. 
Flow from a Tank
Flow under a Sluice Gate
Flow through a Nozzle 

In the Conservation of Energy section, it was shown that for a control volume, the energy equation can be simplified to
In many cases, the head loss (mainly due to viscous effects) can be ignored. If there is no pump or turbine in the system, then the equation becomes
This relationship is a form of the Bernoulli's equation. The same relationship, but in a slightly different form, can be derived by applying
conservation of momentum to a fluid element along any streamline in the flow, giving
p + ρV^{2}/2 + ρgz = constant along streamline
where p is the static pressure, ρV^{2}/2 is the dynamic pressure, and ρgz is
the
hydrostatic pressure.
Bernoulli's equation provides the relationship between pressure, velocity and elevation along a streamline. It can be applied to solve simple problems, such as flow from a tank (free jets), flow under a sluice gate and flow through a nozzle. Applying Bernoulli's equation between points 1 and 2 as shown in the figures yields,
However, one should realize that Bernoulli's equation is subject
to some restrictions, and can only be applied to certain flow situations.
The assumptions made in deriving Bernoulli's equation are:
(1) Steady flow
(2) Incompressible flow
(3) Inviscid flow (zero viscosity)
(4) Flow along
a streamline 
Piezometer and Pitot Tube
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A combination of piezometer and pitot tube
can be used to obtain the velocity at a specific point. Static pressure
can be measured using a piezometer. The Pitot tube, as shown in the figure,
can be used to measure the stagnation
pressure. Stagnation pressure is the pressure when the flow has a stagnation
velocity (i.e., V = 0). By perfectly aligning the Pitot tube with the
flow, the flow will come to a stop at the tip of the Pitot tube, hence
providing the stagnation pressure measurement.
Applying
Bernoulli's equation between points 1 and 2 as shown in the figure,
and canceling the elevations (equal values) gives, 
Stagnation Point
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Since V_{2} = 0, the velocity at point 1 is
where p_{1} and p_{2} are known from the height of the fluid column in the
piezometer and Pitot tube, respectively. 
Energy and Hydraulic
Grade Lines


The
energy grade line (EGL) and the hydraulic grade line (HGL) provide a
graphical interpretation of Bernoulli's equation. The EGL represents
the total head available with respect to a chosen datum (i.e., a reference
line, as shown in the figure).
The EGL is a constant for frictionless flow where no work or heat is
associated with the process. On the other hand, the HGL is the sum of
static pressure
and elevation head.
Sometimes, this is also referred as the piezometric head and is the
height a fluid column would rise in a piezometer. For example, the
EGL and HGL for frictionless flow in a duct are shown in
the figure. 