Is it sensible to harness wind energy on an airplane?
Actually the idea is to build an electric aircraft whose power comes from solar and from wind while the plane is landing. instead of using spoilers as airbrakes, these wind turbines separately mounted could add drag and harness energy at the same time . In other words, regenerative braking.
Normally not: Extracting energy from the airflow produces drag, which must be overcome by added thrust. Since every form of energy conversion produces losses, more thrust energy must be added than can be gained from the airflow.
Only when the engines fail and the generators stop running does it make sense to extract energy from the airflow. In airplanes there are two applications which are driven by "wind energy":
Deployed RAT (picture source)
Rocket-powered airplanes have no easy way to generate electricity, so the Me-163 B used a small windmill at the tip of the fuselage to drive a generator.
Me-163 B (picture source)
EDIT: With your question's new focus on electric propulsion the answer will become different. Now you most likely will have propellers which are powered by electric motors. During landing, those could run inverted and charge a battery, which will most likely be empty by the time the aircraft lands. This can be done during final approach until the end of the runout after touchdown. It can be expected that any electric airplane will have a high L/D, so deploying speed brakes to allow for a steeper approach makes sense.
I would be surprised if an additional device would be economical, however. This recharging must be done by the regular propulsion system, or it would add dead weight during most of the flight.
You asked for formulas, but all I can provide here are some back-of-the-envelope calculations. First, it must be said that variable-pitch propellers will be lousy as windmills, because their camber and twist are wrong for windmilling mode. I would expect that their efficiency is around 30%, which means that only 30% of the energy extracted by drag will be converted into mechanical energy which drives the electric motor.
Next, running an electric motor as a generator will again require compromises. Good motors make poor generators, and re-wiring the motor for better generator performance will degrade its efficiency in normal use. You will quickly lose more than you gain from the short flight phase when running the motor inverted makes sense.
Now let's assume that you keep your propulsion system at top efficiency (say, 90%) and accept that it will only convert 10% of the drag energy into electric energy. Let's also assume that your prospective solar airplane has an L/D of 30 which needs to be reduced to 10 for a practical approach. You do this from 1000 ft downwards and use the windmilling propeller also during the runout. Approach speed is $v$, mass is $m$ and the initial energy of the airplane is $305\cdot m\cdot 9.81 + \frac{m}{2} \cdot v^2$. Two thirds of the potential energy go into propeller drag, and to be generous we assume that 100% of the kinetic energy will also go into propeller drag, even though the braking force of the propellers at low speed is really lousy and needs support from wheel brakes.
Now it is important how fast your airplane flies, because this will shift the ratio between potential and kinetic energy. To keep things simple, I will relate both to the energy required for the next flight. 10% of the full kinetic energy will accelerate the aircraft to less than one third of its flight speed - after this, the remaining 91% of the energy to reach $v$ must be added by charging the batteries between flights.
The electric energy taken from the potential energy will help you climb to 60ft or sustain level flight at $v$ for a distance of 1800ft. At an L/D of 30 the airplane will fly a distance of 30,000ft without thrust, and by braking you will extract the energy to cover 20,000ft, which at 10% conversion efficiency (and 90% propulsion efficiency!) will carry you over just 1800 ft.
Most other answers focus on normal flight; your (updated) question specifically asks about regenerative braking. In theory, yes, it's possible, in practice, no, it's really not a practical idea.
Let's first focus on the descent from cruise altitude to final approach. Ideally, this descent is done with engines on their lowest thrust setting (flight idle), which means the engines are providing power to the various electric and hydraulic systems, and pressurization, and a little bit of thrust since you can't really avoid that on a jet. Theoretically, this could be done by a Ram Air Turbine (let's for now assume that our hypothetical airplane is equipped with a rather big version that comfortably powers all aforementioned systems, since a typical RAT can only provide limited emergency power). However, this will greatly increase the drag, and to maintain speed, the descent profile will be steeper. This in turn means that the airplane must spend a longer time at cruise altitude, which requires energy to maintain.
Another way to look at that is by a simple energy balance: there is a finite amount of potential energy, and it doesn't really matter whether you use it to maintain speed only, or whether you use it to maintain speed and for regenerative braking, since the total amount of energy you can extract from it is always the same.
Of course, there is one moment that airplanes must brake, and this is after touchdown. Let's for now assume that we are very environmentally aware, and do not use any other means of braking than a regenerative braking system. The airplane is still going at about, say, 130kts, 240km/h or 150mph, so surely that will give us a lot of energy? Let's make a back-of-the-envelope calculation for some figures I found for a Boeing 737-300.
Let's say we land weighing 60 000kgs (close to MLW) with flaps reduced to 15, which gives us a landing speed of 158kts=81,3m/s. The amount of kinetic energy is then $\frac{1}{2}mv^2\approx200 MJ$ (yes, that's mega-joules!). That's a lot of energy, right?! Well... not really. Kerosene (which is approximately the same as jet fuel) comes at an energy density of around 46MJ/kg. We're talking about the equivalent of a little over 4kg of kerosene, on an aircraft that carries about 16,000kg of the stuff. That means we're regenerating about 0.025% of the fuel capacity.
I will leave it as an exercise to the reader to think whether a regenerative braking system could be designed, such that the extra fuel used due to its weight and size is under 0.025%.
Edit Let's redo the calculation on an airplane that actually has most of the necessary equipment (batteries and electric motors that could perhaps double as alternators or dynamos) on board: the Solar Impulse 2. It has a whopping 4x41kWh battery capacity (590MJ). Assuming the takeoff and landing speed are the same (20kts=36km/h=10m/s) and with a loaded weight of 2300kg, the kinetic energy upon landing is 115kJ. This is 0.0195% of the battery capacity - about the same as our B733 example! And note that this is again assuming that the propellers recover 100% of the kinetic energy... This idea is never going to work. (For comparison, it's less than two seconds of energy generated by the solar panels at their respective peak rating)
That means we're regenerating about 0.025% of the fuel capacity. That right there is the #1 reason this hasn't and won't be implemented. It's simply not worth it in terms of dollars, economics, or environmental concerns (don't forget it takes natural resources to produce all those regenerative braking components & especially batteries).
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No, because, from the point of view of a flying aircraft, wind does not have any energy.
Wind turbines stand on the ground and the air mass is moving at some velocity past them, so it has kinetic energy. But aircraft is moving relative to the wind, so the wind is the rest frame and has no energy. So when the aircraft is using the air flow and/or ram air pressure, it is using its energy, not the wind's¹.
Now of course a turbine mounted on aircraft will produce energy. But it will be at expense of the aircraft energy. When flying level under power, the power comes from the engines, so it is more efficient to extract it directly via generator mounted on the accessory drive. But even when the aircraft is descending on idle, it will be at expense of its potential energy and that was originally provided by the engines during climb. Shutting down the engine earlier and using all of it to compensate for drag during the glide is more efficient.
This applies to a solar powered motor glider just like any other plane. It is more efficient to stop the engines earlier and glide at around best glide angle than running the engines longer and then regenerating the energy, because neither conversion from electric energy to potential energy via engine and propeller, nor the conversion from potential energy to electric energy via turbine and generator is particularly efficient.
And it applies even when using thermals or other raising air to gain potential energy. It is again more efficient to avoid converting the energy and simply use the thermal to gain altitude and extend the glide while taking electric energy directly from the solar panels.
Also, weight will be important limiting factor for solar motor glider. This means you won't be able to put many batteries on board and that in turn means that regenerative braking won't be very useful as you won't have much capacity to store the energy. It also means you might want to avoid dedicated turbine to save weight; you'll still be able to regenerate some energy from windmilling propellers at somewhat lower efficiency. But as explained, it does not make all that much sense.
The only time aircraft use turbines is during emergency. When all engines fail, ram air turbine is used to power the essential electric and hydraulic systems. It shortens the gliding distance a bit, but it is worth it if no other power sources are available.
As far as wind goes, the only practical use of wind is choosing route so that there is as much tail wind during cruise as possible. For example the North Atlantic Tracks are periodically adjusted to allow the eastbound flights to utilize the Jet Stream. In a sense, this utilizes the wind energy, because the aircraft thus burns less fuel to get to its destination.
¹ Energy is a weird quantity. It is conserved in all inertial reference frames, but some of its forms will have different values in each. You can choose a reference frame where wind does have energy, but it will make less sense.
For the reasons other answerers have said, it isn't normally worthwhile, because it isn't really wind energy, it's the aircraft's energy.
One situation where it is worthwhile is to generate electricity when it is difficult to take energy from the engine directly. For example, the vintage aircraft I fly didn't originally have an electrical system. To power their radios and transponders (essential for practical flight nowadays), they've been retrofitted with a small turbine under the nose. It increases the drag of the aircraft slightly, but the alternative would be trying to fit an alternator to a vintage engine, or replacing the engine completely, both much bigger modifications.
I've also seen touring aircraft which use small wind generators when they're tied down on the airfield, presumably to keep the battery topped up. The generator in that case is a small wind turbine atop a vertical pole, similar to what you might see on a houseboat or a caravan. They're removed and stowed before flight, so they're not quite what you're thinking of, but they're still aircraft using wind power.
I think no one really adressed your question. I'll try to, as far as I understood. (I still wonder why you need a plane landed with non-empty batteries, but let's assume it's for faster refuel & takeoff?)
First of all, it is only possible to harvest wind energy (of air moving w.r.t. ground) if you have access to the ground.(You need to be a the interface between the two moving objects). A wind turbine must be grounded, a sail boat is in contact with the water, etc. For an aircraft this is not the case obviously, so as said in other answers it is impossible to gather free wind energy mid-flight.
However, it is possible to use windspeed to harvest some of the aircraft's energy. Let us recall there are three main energy tanks in an aircraft (i'll use A320 and TB20 (sorry french link for numbers) both in cruise in a regular mission, it would scale nicely for any e-aircraft):
As you can see, fuel energy in a classic airplane really dwarfs potential energy about 20-to-1, and potential surpasses kinetic about 10-to-1. It's the travelling part that costs the most, and unfortunately, airplanes are kinda designed expressedly for travel :D
Since what you used to travel (against the drag) can never be recovered, only the surplus kinetic and potential at the end of the mission may be. You propose to perform the descent and landing with the wind turbines. Let's suppose you have the wind turbine for the job. You may only re-gather kinetic energy during approach and landing, and potential energy during descent at constant-speed. Even if you manage to recover the whole energy, you only get ~5% of what you used in your whole mission! (and i used 100% efficiency everywhere, full system should be close to 20% efficient including turbine, generators, power unit, batteries etc. so we're really speaking about 1% net energy return).
That said, there might be a use for a windspeed turbine in very specific missions, but you would prob. drop the solar panels. I'm thinking parachuting planes. Mission: go high very quickly, drop your buddies, descend, repeat.
For this the energy demand isn't very high since you're not travelling. I'm thinking a mission requirement of 2x pot. energy to:
I'm saying drop the solar panels, because this will be very power-intensive, and solar panels just have too low energy density
Note: energy density of a system is energy packed/mass on board. For batteries and fuel this is straight forward, for solar panels it's different: the longer the mission, the more energy produced, the higher the power density. This mission is so short it's not worth it.
TL;DR : Unless you have a very weird mission, there just isn't enough energy (~1%) to collect to be worth bringing a few wind turbines for a whole travelling mission.
Airplanes move pretty fast (in some cases very fast) and have lots of drag. So they use lots of energy just to cruise.
Regenerative braking allows you to capture, as an absolute upper limit, all the kinetic energy and gravitational potential energy that the plane has when it starts its descent (obviously there's then an efficiency consideration too, so it will actually be less than this). The work you've done during the flight just to overcome drag at cruising speed is gone no matter what. So first thing: any drag that your doohickey adds to the plane while it's not braking, is costing energy through the whole flight. Let's assume that it's somehow stowed away, just like landing gear can be, and figure that it's going to bulk the plane up a bit but it isn't going to totally ruin the aerodynamics.
Now, on a typical flight what proportion of the fuel used, is used to reach cruising altitude and speed? I don't actually know, and of course it depends on the length of the flight among other things, but I'm sure some actual pilots could chip in with rough figures.
The energy to get up is an upper limit (again, there's inefficiency in the engines) on how much energy you can extract from the process of getting down. So, that proportion of fuel used to get up puts an absolute upper limit on the proportion by which regenerative braking can possibly reduce the total energy requirements of your solar-powered plane. And that's before considering inefficiencies of both processes (the engines and the regenerative brakes). So to completely invent some figures, supposing take-off is 25% of fuel for some particular journey, and combined efficiency is 50%, then regenerative braking could perhaps reduce energy requirements by 12.5%. This seems worthwhile on the face of it, but (a) I've picked what I think are overly-large numbers, and (b) we haven't yet paid for the mechanism that does it.
Can the batteries (or whatever) that store the energy from the brakes, store more energy than it costs to add them to the plane, thereby hauling a larger weight up to altitude in the first place, and the energy lost in cruise to the extra drag imposed by incorporating the whole system? In the worst case where they can't, you've made a net loss by adding regenerative braking.
Regenerative braking works reasonably well for cars, especially in city driving, because they slow down frequently and so they're otherwise dispersing a lot of unwanted kinetic energy as heat. To a rough approximation, planes only slow down once per trip. And I believe it's harder to efficiently capture the energy when you're braking against air rather than against static friction on the road, and so the efficiency of the system on the plane will be much less than that for a car. So I don't think your plan is in good shape right now :-)
The normal way to harness the energy of the air in an aircraft is a glider.
The wikipedia article covers it pretty well: find an area where the air is rising, and use it to lift the aircraft up. There is no fancy energy conversion on the aircraft, it's all done directly by the lifting surfaces.
Solar-powered planes have been built, although the necessary compromises mean they're not yet popular. Some tech companies are looking at the possibility of high-endurance automated solar aircraft as radio relays for connecting remote areas to the Internet.
Very Short Answer
Is it sensible to harness wind energy on an airplane?
I assume as "wind energy" you mean, extracting energy from the "airflow". In theory yes, only in descent, if you can prove a design which brings benefits in terms of energy, complexity, economics. In practice, you have very slim margins with your solution, and only "one case" when harvesting energy makes sense.
Little Longer Answer
Let's analyze the idea:
... to build an electric aircraft whose power comes from solar and from wind while the plane is landing.
While the airplane is landing it is descending from "cruise" altitude to the airport. In theory it would need no "extra power", since you have a lot of potential energy, you could just use this potential energy to "glide" to the airport, similar on what sailplanes do. An important note the energy to get to "cruise" altitude has already been spent, for example, what Solar Impulse 2 do is to climb and recharge the batteries during the day, and glide and use the energy-from-the-batteries during night. Don't underestimate the efficiency of the gliding maneuver over night, with altitude you gain a lot of potential energy! You would not gain the same energy by harvesting it to from the airflow because you are transforming this potential energy in electrical energy with a transformation chain that bring you some losses.
That said, the one case when makes sense to harvest energy in descent is the moment that you need to loose some altitude in a "short" period of time, (not glide to the ground, which would be the most efficient solution). In this case it is desirable and possible to "harvest" some energy, what you are proposing is:
instead of using spoilers as airbrakes, these wind turbines separately mounted could add drag and harness energy at the same time . In other words, regenerative braking.
This is correct, it is done in a similar way with Ram Air turbines in emergency situations (no power) on larger jets. On a variable-pitch propeller aircraft theoretically you could change the pitch of the propeller to make them work as windmills, so that you would not need to add extra complexity and weight. If you have an electric motor driving the propellers you could potentially using it as a generator. In case you add "wind turbines", or any extra system, you should make sure that:
A last note: Spoilers are use not only to add drag but even to reduce lift, the main use case here is descent (emergency or not)!
Aircraft generate lift by increasing airspeed, while wind harvesters generate power by decreasing airspeed, so they'd be competing with each other.
Assuming that there was a way to avoid that problem, it'd also need to store this energy, and large capacity batteries are incompatible with an aircraft's requirements for both high specific energy and high specific power.
My company has just secured a patent for an Energy Cell Regenerative System For Electrically Powered Aircraft (May 9, 2017).
A properly designed and efficient regen system can flip the prohibitive physics enough for a net gain and be a worthwhile addition to an electric aircraft. A key is to design a lower pressure area beneath a cowling venturi and harness some of the prop blast in flight (there is some) it tornadoes around the fuselage but beneath the cowling is,combined with the speed of the aircraft, a source of kinetic energy.
The turbine is a centrifugal design and is tucked away under a blister which makes the system nearly drag free when not in use. The PMA can produce 3kw and weighs 19 pounds. The whole system weighs 31 pounds and includes a mosfet controller and a buck boost inverter similar in design to those found on Toyota's Prius. There is a lightweight Maxwell super capacitor module as a "charger". Power generated from the system charges a probably lithium battery pack that has been designed to be charged and discharged simultaneously (see the patent), through an algorithm managed by the controller. The lighter we can produce the system and also speed up charging by reducing the battery's internal resistance the more efficient, and practical, it will be.
We haven't been permitted to flight test it, but calculations derived from a Cessna 152 mule indicate good potential for regen. It can be used as a rescue system, during descent, or when practical to adjust prop pitch. Knee jerk reaction from laymen and engineers has been negative until they carefully study the invention, then consensus seems to be "That should definitely be developed further". Patent is on the market. Patent #US 9,643,729 B2
This will be built-in, and almost mandatory.
Your question has a built-in assumption: regenerating energy is useful. The only way it would be useful is if the propulsion system (or part of it) was electric. Therefore by definition we must be referring to an aircraft with an electric propulsion.
Also presumably, we are not throwing 1880-era tech into the sky. That means we are using AC motors with flux vector drive, i.e. Large semiconductors synthesizing 3-phase sinewaves from DC.
If the drive frequency is the same as motor speed, that is inert - no current flows and no thrust is made. If a higher frequency, this applies power. If higher still, it applies more power. If slower, it regenerates. If slowerer, it regens more.
So if you have an AC electric drive, you already have regen. Done!
It'll essentially be thrust reverse. Where do you get to use that? Glide slopes are designed very shallow so you'll usually need power all the way down. Look at how they fly the C-17, they would need to redesign approaches to be like that, which would also make them exclusive to electrics and thrust-reverse-in-flight machines like the C-17.
