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Bernoulli's equation is a simple but incredibly
important equation in physics and engineering
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that can help us understand a lot about the flow
of fluids in the world around us. It essentially
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describes the relationship between the pressure,
velocity and elevation of a flowing fluid.
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It has countless applications. We can use
it to explain how planes generate lift,
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or to calculate how fast liquid will
drain from a container, for example.
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We'll explore these applications and a
few more later on, but let's start by
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reviewing the equation itself.
It was first published by the Swiss
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physicist Daniel Bernoulli in
1738, and it looks like this.
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The equation states that the sum of these three
terms remains constant along a streamline.
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Each of the terms is a pressure.
The first term is the static pressure,
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which is just the pressure P of the fluid.
Then we have the dynamic pressure
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which is a function of the fluid density
Rho and velocity V, and represents the
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fluid kinetic energy per unit volume.
And the last term is the hydrostatic pressure,
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which is the pressure exerted by the fluid due
to gravity. G is gravitational acceleration
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and H is the elevation of the fluid, which is
just its height above a reference level.
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This is the pressure form of the equation,
but it can also be presented in the head form,
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and the energy form.
We can think of Bernoulli's
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equation as a statement of the conservation
of energy. It says that along a streamline
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the sum of the pressure energy, kinetic energy
and potential energy remains constant. This is
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really valuable information that can help us
analyse a whole range of fluid flow problems.
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The equation does have a few limitations,
which I'll cover later on in the video,
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but for now the important thing to note is
that it can only be applied along a streamline.
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We can define a streamline in steady flow as the
path traced by a single particle within the fluid.
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Or more technically as a curve that at all points
is tangent to the particle velocity vector.
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Let's look at an example where
we apply Bernoulli's equation
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to flow through a pipe which has a change in
diameter. We want to use the equation to see
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how the pressure changes as the flow passes
from the larger to the smaller diameter.
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Bernoulli's equation is usually used to
compare the flow at two different locations,
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so we can rewrite it like this, with points
1 and 2 both being on the same streamline.
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There’s no significant change in
elevation between Points 1 and 2,
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so the potential energy terms cancel each
other out. And if we put all of the static
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pressure terms on one side we get this
equation for the change in pressure.
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If we assume that the fluid is incompressible,
the mass flow rate at points 1 and 2
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must be equal. This gives us what’s called the
continuity equation, which is just a statement
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of the conservation of mass. Mass flow rate
is equal to the product of the fluid density,
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the pipe cross-sectional
area and the fluid velocity.
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So we can re-arrange the continuity equation to
obtain an equation for the velocity at point 2.
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The cross-sectional area A2 is smaller than
A1, which means that the velocity of the
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flow increases as it passes into the smaller
diameter pipe. This is quite intuitive.
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By substituting this equation for V2 into
Bernoulli's equation, we can see that since the
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velocity increases between Points 1 and 2, the
pressure between both points must decrease.
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This concept, that for horizontal flow an
increase in fluid velocity must be accompanied
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by a decrease in pressure, is one way of
formulating what we call Bernoulli's Principle.
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It can seem counter-intuitive,
because people often expect an
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increase in velocity to result in a
corresponding increase in pressure.
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But it makes sense if we think about the
conservation of energy. The energy required
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to increase the fluid velocity comes at the
expense of the static pressure energy.
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Bernoulli’s Principle shows up
in a lot of different places.
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We can use it to help explain how plane
wings generate lift. Fluid flowing over
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an airfoil travels faster
than fluid flowing below it.
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According to Bernoulli's Principle this creates
an area of low pressure above the airfoil and
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an area of high pressure below it, and it’s
this pressure difference that generates lift.
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I'll cover lift and drag forces in
more detail in a separate video.
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Bernoulli's Principle also explains
how Bunsen burners work.
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When the gas valve is opened, gas flows into the
barrel at high velocity. Following Bernoulli’s
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Principle, this high velocity creates an area
of low pressure in the barrel, which draws
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air in through the air regulator, allowing
for more complete combustion of the gas.
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Several different flow measurement
devices rely on Bernoulli’s equation
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to determine the velocity of a flowing fluid.
The Pitot-static tube is one such device.
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It’s often used in aircraft to measure
airspeed. Here’s how it works.
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If we place a tube into a flowing fluid,
like this, and we attach a pressure meter
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to the end of it, the meter will measure
the pressure at the end of the tube.
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At this point the fluid velocity is reduced
to zero, so it’s called the stagnation point,
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and the pressure measured by the meter
is called the stagnation pressure.
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We can apply Bernoulli’s equation between
an upstream point and the stagnation point,
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and show that the stagnation pressure is
equal to the sum of the static pressure
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and the dynamic pressure terms. All of the
kinetic energy is essentially being converted
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into pressure energy at the stagnation point.
If we add an outer tube which is sealed at the end
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but has holes further downstream, the outer tube
will measure the static pressure of the fluid,
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instead of the stagnation pressure.
These two pressure measurements give
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us all of the information we need to
determine the velocity of the flow.
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Another flow measurement device
that uses Bernoulli’s equation
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is the Venturi meter, which is an instrument
used to determine the flowrate through a pipe.
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It works by measuring the pressure drop
across a converging section of the pipe.
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Say we want to determine the flow rate Q,
which is the velocity multiplied by the
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pipe cross-sectional area at Point 1. We can
easily rearrange the pressure drop equation
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we derived earlier when we looked at a change
in diameter, to get this equation for flowrate.
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All we need to know is the
dimensions of the Venturi meter,
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the fluid density and the pressures P1 and P2,
and that allows us to calculate the flowrate.
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The Venturi meter has no moving parts
and is a very simple and reliable way
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of measuring the flowrate through a pipe.
The diverging section is longer than the
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converging section to reduce the likelihood of
flow separation and keep energy losses low.
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Let's look at one more example where
we can apply Bernoulli's equation.
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Say we have a beer keg, and we want to
calculate how fast will drain when we
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first open the tap at the bottom.
All we need to do is define our two
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points along a streamline and
apply Bernoulli's equation.
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It’s a gravity-fed keg with a vent at the top,
meaning that it’s not pressurised. The pressure
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at both points will be atmospheric, and so the
static pressure terms cancel each other out.
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We can also assume that the keg
is large enough that the fluid
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velocity at Point 1 is close to zero. If we rearrange Bernoulli’s equation,
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and define the height between
the beer level and the tap as H,
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we get this equation for the
beer velocity out of the tap.
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Those were a few examples of cases where we
can apply Bernoulli's equation to get some
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valuable information or to solve a problem. But to use it correctly, it’s important to have an
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understanding of the limitations of the equation,
which arise because of how it’s derived.
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There are several different ways
Bernoulli’s equation can be derived.
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It can be derived based on conservation of
energy, by considering that the work done
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on the fluid increases its kinetic energy.
Or it can be derived by applying Newton's second
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law, which involves determining the forces acting
on a fluid particle and applying F equals M*A.
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Although I won't cover either derivation
here, they do both make some assumptions
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that we need to be aware of, since they
limit how we can apply the equation.
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Firstly the derivation of Bernoulli’s equation
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assumes that flow is laminar and that it is
steady, meaning that it doesn't vary with time.
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Next, it assumes that the flow is inviscid,
meaning that shear forces due to fluid
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viscosity are negligible. This assumption
is needed because viscosity would result
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in a dissipation of some of the fluid’s internal
energy, and so the idea that energy is conserved
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along a streamline would no longer apply.
And finally the derivation of Bernoulli's
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equation assumes that the fluid behaves as if it’s
incompressible. This is usually valid for liquids,
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but might not be for gases at high velocities.
All three of these assumptions need to be valid if
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you want to apply Bernoulli's equation.
Adapted versions of the equation which can
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be applied to unsteady and compressible flows do
exist, although they’re a bit more complicated.
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Being able to recognise when Bernoulli’s
Principle is at play, or when Bernoulli’s
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equation can be applied to solve a problem,
is a powerful tool in any engineer's arsenal.
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If you'd like to see a few more real world
examples of Bernoulli’s principle in action,
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