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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.
If you'd like to see a few more real world examples of Bernoulli’s principle in action,
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