The world isn’t perfect.

No matter how hard you try to solve a problem, or prevent one from happening, things still go wrong. That’s just how life is.

And sometimes, that means your basement gets flooded. So, what do you do? You need to get the water out, but how?

Well, you’re going to need some equipment. Some pipes. A pump.

And to choose the right pump for the job, you’ll need Bernoulli’s Principle. [Theme Music] If you’ve seen our last episode, you already know all about fluid mechanics and the energy transfers involved. But engineering is about more than just learning – it’s also about using your knowledge to build things and solve problems. Like your flooded basement.

To get the water out, you’ll need a pump – a device that’s used to move liquids, compress gases, or force air into things like tires. In this case, obviously, you’ll want a pump that’s designed to move liquid, and powerful enough to push all that water in the basement somewhere else, like outside. Preferably far away.

You could just pick a random pump and hope it works, but that’s not the engineering way. To do this properly, you can calculate how powerful your pump needs to be with the help of 18th-century Swiss mathematician Daniel Bernoulli. There were actually eight mathematicians in the Bernoulli family in the 17th and 18th centuries, including Daniel’s father and two brothers.

His father was especially jealous of his success. Their rivalry was so bad that when the two of them jointly won a scientific prize, his father banned him from the house. He also went on to plagiarize Daniel’s later work, changing the date so it seemed like he’d actually published it before his son.

As competitive as their relationship was, it’s possible that the challenge pushed the younger Bernoulli to become a better innovator and mathematician. But either way, he discovered a good deal about fluid flow in his life. He specifically wanted to understand the relationship between the speed at which blood flows and its pressure.

So to learn more, he conducted an experiment on a pipe filled with fluid. Bernoulli noticed that when he punctured the wall of the pipe with an open-ended straw, the height to which the fluid rose in the straw was related to the pressure of the fluid in the pipe. Soon, physicians all over Europe were measuring their patients’ blood pressure by sticking sharp glass tubes directly into their arteries.

Luckily for us, they’ve developed more gentle methods since. But while the physicians of his day were being quite the pain in the arm, Bernoulli was on to something very important. He realized that energy was conserved in a moving fluid.

It could be converted between different forms, like kinetic energy – the energy of motion – and potential energy, but the total energy within the fluid would stay the same. So if one form of energy decreases, for example, like if the fluid slows down, there has to be a corresponding increase in another form of energy so the total remains constant. Bernoulli’s insight was that energy could also be converted between kinetic and potential energy and pressure.

Today, this is known as Bernoulli’s Principle, and it says that as the speed of fluid flowing horizontally increases, the pressure drop will decrease, and vice versa. This means that the fluid’s speed will have an inverse relationship with its pressure, or that as one rises, the other falls. Now, this principle only really applies to what’s known as an isentropic flow, meaning it doesn’t involve any heat transfer, and it’s reversible, so it can go back to its initial state with no outside work.

Or at least, close enough that you can neglect the effects of heat transfer or irreversibility. To keep things simple, we’ll assume this applies to our system. For Bernoulli’s Principle, the fluid also needs to flow horizontally, or not have a drastic change in height, because it doesn’t consider the effects of gravity.

So to account for height and gravity, and apply Bernoulli’s Principle to the design of our pump system, we’re going to need a more general equation. Bernoulli’s Equation. There are many forms of Bernoulli’s Equation, but the one we’ll look at today relates the pressure, speed, and height of any two points in a steadily flowing fluid with a density ρ.

Since something’s density is just its mass divided by its volume, this equation actually works out really neatly: On both sides of the equation, you’ll see that we’re defining a point’s total energy per unit volume by its pressure, plus its kinetic energy per unit volume, and then finally adding in its potential energy per unit volume. Basically, this equation says that the total energy of the first point is equal to the total energy of the second point. So if, say, the two points have different speeds, it makes sense that they’d have different pressures or potential energy to balance out the equation.

Since the total energy will be the same at every point in the fluid, you can also write the equation like this. It’s very similar to what we had before, but since the total energy will always be the same, you can replace the right side of the equation – which represents the total energy at a second point along the pipe – with a constant. Much simpler.

And it gets even simpler if the fluid only flows horizontally – in other words, if there’s no change in height between the points we’re comparing. That means we can cancel out the term with potential energy, leaving us with only the pressure and the kinetic energy per unit volume equal to a constant. By now, you’re probably wondering how we can find out the actual value of this constant we keep talking about.

Well, we know it’s the total energy per unit volume, and that one way to find it is to add up the pressure, kinetic energy over volume, and potential energy over volume. But when we’re talking about a real-life scenario with a pump involved, we have to take into account the energy that’s being put into and lost from the system to move the fluid. We’ve said this energy can take two forms: work and heat.

Work is what’s driving the pump. That work is what’s moving the water along, and contributing to the total energy. But it’s not the only factor.

There’s something else that we need to take into account: friction. As water flows through the pipe, the movement will induce stress in the fluid, which causes friction – the resistance you get when two things slide against each other. Friction makes a system lose energy to heat, and there’s going to be a lot of it as the water rubs against the inside of the pipe.

Now, even with the pump, the total energy of the fluid will still be constant throughout the pipe. Which means the changes in energy from work and friction will need to balance out with changes to the three forms of internal energy – so, pressure, kinetic energy, and potential energy. That’s the only way the total energy will remain the same.

Going back to Bernoulli’s Equation, we can now modify it slightly. The first thing we’ll do is rewrite it in terms of changes in energy: On the left side, there are the changes to the internal forms of energy: the change in pressure, plus the change in kinetic energy per unit volume, plus the change in potential energy per unit volume. That’s all equal to a constant.

Then, we’ll divide the whole thing by density, which remember, is really the same as multiplying by volume and dividing by mass. So now we’re talking about overall changes per unit mass throughout the flow, instead of changes per unit volume. This is what needs to balance out with the energy added by the pump and lost to friction.

So, instead of a constant, we can now say the left side of the equation is equal to W, the work put in to change the energy per unit mass, minus frictional losses per unit mass. Many engineers will take this equation a step further and divide the whole thing by gravity, which gets you the value known as head – the height to which a pump can drive the fluid. Now, the actual amount of energy lost to friction depends on a bunch of different parameters.

One of the big ones is the velocity of the fluid. The greater its velocity, the greater the frictional losses. It makes sense if you think about it – there’s going to be much more intense rubbing against the sides of the pipe if the water’s moving faster.

And remember last episode when we talked about laminar and turbulent flow? Well that matters here too, because a turbulent, or fluctuating flow will increase the friction more than a laminar, or smooth flow will. The length of the pipe matters, too.

If there’s more pipe for the water to rub against, it will lose more energy to friction. And then there’s roughness: the rougher the pipe, the more friction there will be. Not to mention every valve, fitting, bend, and intersection in the pipe, which will also increase the friction.

To figure out how powerful the pump needs to be to get the water out of your basement – in other words, how much work it should be capable of producing – you’ll have to account for all of this. You’ll want to minimize friction by keeping the pipe as simple and short as possible – this way, you won’t need as much work to counter the energy loss. The pump also needs to be able to perform enough work to account for the pressure and velocity of the water, as well as any changes in elevation.

Reality tends to be a little messier than the simplified version of Bernoulli’s equation we’re using, but it should be enough to get a sense of which pump you’ll need to finally clear out your basement. Who knows how much mold is growing down there by now. More generally, Bernoulli’s equation is a good foundation for working with fluids and figuring out how to build your designs around them.

The world isn’t always perfect, but with the right engineering skills and tools, it doesn't have to be. So today was all about diving further into fluid flow and how we can use equipment to apply our skills. We talked about Bernoulli’s Principle and the relationship between speed and pressure in certain flowing fluids.

We then learned how to apply the principle with Bernoulli’s Equation. Taking that equation, and substituting a constant with work and frictional loss, gave us a great way to use it in real-world examples. I’ll see you next time, when we’ll talk all about heat transfer.

Make sure you bring a bottle of sunscreen. Crash Course Engineering is produced in association with PBS Digital Studios. You can head over to their channel to check out a playlist of their latest amazing shows, like Brain Craft, Deep Look, and PBS Space Time.

Crash Course is a Complexly production and this episode was filmed in the Doctor Cheryl C. Kinney Studio with the help of these wonderful people. And our amazing graphics team is Thought Cafe.