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