Understanding Bernoulli's Equation

Thanks to CuriosityStream for sponsoring  this video. Watch thousands of documentaries and get access to Nebula for free, when you  sign up using the link in the description. 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, you can check out the extended  version of this video on Nebula.

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