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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 itvz. 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 forvz 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

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  • 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,

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  • As always thanks for watching and thank you to all my Patreon supporters. If you would

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This episode of Real Engineering is brought to you by Brilliant, a problem solving website

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Are Electric Planes Possible?

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    joey joey posted on 2021/06/05
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