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
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My eyebrows rose steadily until "don't have enough ridiculous assumptions in this equation", self awareness is important!
Great video as always man.
Wait, wouldn't the comparatively low energy efficiency of combustion engines compared to electric engines eat at least some of the advantage that the fossil fuel has?
Or have I missed tbat?
A plane uses around 1/5 of its fuel during takeoff and ascent. You could probably gain a fair amount of efficiency if you had a two-stage setup. That's a bit drastic for a 20% increase in efficiency though.