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 anotherform 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.