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This episode of Real Engineering is brought to you by Brilliant, a problem solving website
that teaches you think like an engineer.
Over the past decade we have seen multiple industries looking to transition to renewable
fuel sources, and while we have been making huge strides in the production of renewable
energy, the technology required to allow every industry to use it has not kept pace. In theory
we could replace every coal burning power plant in the world in the morning, and manage
just fine, IF we had a reasonable way of storing that energy cost effectively and efficiently.
This energy storage dilemma is slowing our adoption of renewable energy, and one of the
industries this is most apparent is the aviation and aerospace industry. Elon Musk is running
around pushing electric vehicles and solar powered homes, yet every time a Falcon 9 launches
it burns 147 tonnes of fossil fuel. Boeing and Airbus are in a constant battle to create
the most fuel efficient plane, allowing their customers to save on ever increasing fuel
costs and increase their bottom line, yet they are still using kerosene, when energy
from the grid is cheaper.
So what gives? Why isn’t every industry on earth clawing at the prospect of transitioning
to renewable fuels? The aviation industry has one massive hurdle to overcome before
it can successful adopt renewable energy. The energy density of our storage methods.
Energy density is a measure of the energy we can harness from 1 kilogram of an energy
source. For kerosene, the fuel jet airliners use, that’s about 43 MJ/kg. Currently even
our best lithium ion batteries come in around 1 MJ/kg. Battery energy is over 40 times heavier
than jet fuel.
So why is this such a huge problem. A plane flies when lift equals the weight of the plane,
so when we increase the weight, we have to increase the lift, which requires more power.
Needing more power means we need more batteries, which increases the weight again. So are caught
in a catch 22 of design.
We could end the video there, but going by the demographic breakdown of this channel,
we can go a little deeper. To really understand why this is such a difficult problem, let’s
do some back of the envelope calculations to convert two planes, the Airbus a320 and
a small personal aircraft like a Cessna, to battery power. Ultimately, we want to know
the power requirements of flight and how it will draw on the energy supply of the battery.
Animation 5 The work-energy theorem tells us that Work
= F × ∆x, where delta x is the distance over which a force acts. Power is work per
unit time, so P equals work divided time. (Work/∆t). Inserting our equation for work
and we get an equation for power that equals Force multiplied by distance divided by time,
otherwise known as velocity. Here delta v is the velocity of whatever is getting worked
on, in this case it’s the air. When a plane is flying at a constant height, we know that
the the force of lift and the force of gravity are balanced. That means the upward force
of lift (Flift) has to be equal in magnitude to the downward pull of gravity, which equals
the mass of the plane multiplied by gravity So, the power required for lift equals the
mass of the plane multiplied by gravity and delta V.
The question is, what is delta v? It’s the downward velocity of the air that the plane
pushes downward. So let’s call it ∆vz. To find its value, we have to think about
the mechanism of lift.
The lift an airplane provides is equal to the rate it delivers downward momentum to
the air it displaces.This means that the force of gravity must be equal in magnitude to the
downward velocity of the deflected air, times the rate at which air gets deflected:
The mass of air that the plane affects is simply the volume of the cylinder that it
sweeps out per unit time, times the density of air. If we call the relevant cross sectional
area, Asweep, then the volume it sweeps out per unit time is A sweep times the velocity
of the plane. Therefore the mass flow rate equals the density of air times the cross-sectional
area times the velocity of the plane.
Now the only outstanding quantity that we don’t know is the area of air affected by
the plane, Asweep. This isn’t the cross sectional area of the plane, it’s the area
of influence the plane has on the surrounding air. This changes with the relative velocity
of the plane and the air around it, but at cruising speed, the plane dissipates vortices
that have roughly the radius of the length of the plane’s wings. Approximating this
circle as a square because we don’t have enough ridiculous assumptions in this calculation,
the relevant area becomes L squared at cruising speed. Putting it all together, we have the
force lift needs to provide with this equation. This equation is simply telling us the plane
is sweeping out a tube of air and shifting it downwards, and that downward acceleration
of air is equal to the downward pull of gravity on the plane. So the plane avoids falling
by constantly paying the tax of streaming momentum downward via the air.
Rearranging this equation, we can now solve for ∆vz in terms of quantities we can easily
measure. And plugging this into our power equation, the power needed for lift is given
by this equation:
With this equation at hand, we can start noticing what variables really impact the energy requirements
of the plane. Notice that as the plane flies faster the power drawn by the engine actually
gets smaller, but this equation neglects to consider drag. It just so happens, that the
total power needed to fly is minimized when the force of lift and the force of drag become
equal, so we simply to to double our power requirements to get our total power requirement
at cruising speed.
Now we are getting a real picture of why increasing the mass of a plane is such a huge issue.
The mass component of this equation is not only squared, but also doubled. Doubling the
mass will increase our power requirements 8 fold.
With this knowledge in hand, let’s start calculating the real world consequences of
converting an Airbus a32 To start, we can take the battery weight to be the usual mass
fraction that’s devoted to fuel, about 20% of the planes mass for both. We also need
to take into account the fact that at the cruising altitude, the atmosphere is much
thinner than at ground level. For the Cessna, the density falls by factor of 2, and for
the Airbus, a factor of 3. Let’s be generous, and take the specific
power of leading edge Lithium-ion systems, at about 0.340 kilowatts per kilogram kW/kg.
To meet the power demand, the Airbus and would need 31 tonnes of batteries:
10 500 kW / 0.340 kW/kg ≈ 31 000 kg (10)
while the Cessna would need just 100 kilograms:
35 kW / 0.340 kW/kg ≈ 100 kg (11) For the Cessna, this compares very favorably
with the typical weight of fuel it would carry otherwise, and it isn’t terrible for the
Airbus, but this is just the power the plane needs at any one moment in time. What we are
really interested in is the weight of batteries we would need to match the typical range of
these planes.
For the Airbus that’ s a 7 hr flight from JFK to LHR and for a Cessna, that might be
a four hour flight from New York to South Carolina. The energy capacity required for
a trip is given this equation, multiplying the power required for flight by the duration
of the flight:
Again if we use leading edge figures for Lithium ion battery capacity, we can store about 278
watt hours per kilogram.
For the Cessna, the equivalent battery weight is around 500 kg or just less than two thirds
the weight of the plane without fuel. For the A320, the required battery weight is around
260,000 250 000 kilograms or about 4 times the weight of the empty airplane! Compared
to the typical 20% that’s allocated to fuel, this is devastating.
Now that we have a base figure for how heavy the batteries are going to be, we can re-calculate
the actual range taking the added weight of the batteries into account. Let’s assume
that at the very least, we’re not going to accept reduction in flight speed or increases
in total energy used per flight. How much is the range diminished for flights of similar
speed and total energy?
As expected, this downgrades the Cessna’s flight time from 4 hr to about 2 hr. Not negligible,
but livable. A two seater Cessna usually holds about 150 kg fuel and another 100 kg for a
passengers and luggage. It is easy to imagine endowing the Cessna with the required battery
capacity through a combination of lowering the carrying capacity, lowering speed, increasing
wingspan, with lighter parts and more efficient electric engines. In fact, this is exactly
what we are seeing with small electric aircraft coming to market in the past few years, like
the Alpha Electro.
However, the downgrade is substantial for the a320, taking us from 7 hours down to just
20 min, less than one twentieth of the way across the Atlantic.
If we plot the flight duration as a function of battery mass for both planes, we can see
that the Cessna is already sitting around the optimum and could actually increase our
battery capacity and improve our flight range. It’s a different story for the airbus, where
we overshot our optimum battery capacity significantly. Reducing our battery weight to 60 tonnes will
increase our flight duration by about 15 minutes. So we could last a little bit longer before
crashing into the ocean, assuming we could find a place to fit those 60 tonnes of batteries
in the first place.
But we have been seeing great strides with short range small aircraft coming to market,
and if we fly very slowly with low drag wings we can even build a solar powered drone that
never has to land. We won’t be seeing airliners using electric engines any time soon, unless
we can find a more energy dense medium for storing that energy. We will be exploring
one such possibility in our next video. The Truth about Hydrogen.
The derivation of the equations in this video may seem a little difficult to new comers,
but if you follow along you will see it’s just taking basic known equations and combining
them until we have a new equation that solves problem. The value this skill will give you
is immeasurable, but it takes practice. Thankfully Brilliant has a course that allows you to
do just that. Take this course on Algebra through Puzzles. By the end of this course,
you’ll know unique problem-solving approaches in Algebra that aren’t typically covered
in school, and have improved intuition and strategic thinking that you can use when approaching
difficult problems. If you go to Brilliant.org/RealEngineering,
you can do the entire course for free!
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