The Bernoulli equation can be applied to a great many
situations
not just the pipe flow we have been considering
up to now. In the following
sections we will see some
examples of its application to flow measurement from
tanks,
within pipes as well as in open channels.
Flow from a
Tank through a small Orifice
Liquid flows from a tank through a orifice close to
the bottom. The Bernoulli equation can be adapted to a streamline from the surface (1) to the orifice (2) as (e1):
Since (1) and (2)'s heights
from a common reference is related as (e2), and the equation of continuity can be
expressed as (e3), it's possible to transform (e1) to (e4).
Pitot Tube
If a stream of uniform velocity flows into a blunt body, the stream
lines take a pattern similar to this:
From the Bernoulli equation we can calculate the pressure at this point. Apply Bernoulli along the central streamline from a point upstream where the velocity is u1 and the pressure p1 to the stagnation point of the blunt body where the velocity is zero, u2 = 0. Also z1 = z2.
This increase in pressure which bring the fluid to rest is
called the dynamic pressure.
Dynamic pressure = 
or converting this to head (using 
)
)
Dynamic head = 
The total pressure is know as the stagnation
pressure (or total pressure)
Stagnation pressure =
or in terms of head
Stagnation head = 
The blunt body stopping the fluid does not have to be a solid.
I could
be a static column of fluid. Two piezometers, one as
normal and one as a Pitot
tube within the pipe can be used
in an arrangement shown below to measure
velocity of flow.
Using the above theory, we have the equation for p2 ,
We now have an expression for velocity obtained from two
pressure
measurements and the application of the Bernoulli
equation.
Example
The outlet
velocity of a pressurized tank where
p1 = 0.2 (MN/m2)
p2 = 0.1 (MN/m2)
A2 / A1 = 0.01
h = 10 (m)
can be
calculated as
V2 = ( (2 / (1
- (0.01)2) ((0.2 106 N/m2)
- (0.1 106 N/m2))
/ (1000 kg/m3) + (9.81 m/s2) (10 m)))1/2
= 19.9 (m/s)
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