With the increasing interest in the use of multi-rotor UAVs for long range applications such as pipeline inspection, large-area mapping, transporting packages or even people, the question arises: “how far can they fly?”.

In this post I’ll present a theoretical approach, an experiment and a conclusion to answer this question. Let’s start with some basic theory that can be used for all flying things. The distance *d* that a UAV can cover, flying at a constant mass and velocity *v* is presented by:

With *t* the flight time calculated as the available energy in the battery divided by the total required electrical power:

Assuming the power required for propulsion is some orders of magnitude larger then power for payload or avionics, the total required electrical power can be calculated as the power required for propulsion divided by the overall efficiency of the propulsion system (losses in cables, ESC, motors, propellers, air).

For the same battery chemistry and manufacturing process, we can assume that the energy available from the battery linearly depends on its mass:

For Li-Po batteries, a value of 180Wh/kg is possible or in SI units:

The mass of the battery is a fraction of the total mass of the UAV. For example a UAV with a total mass of 4kg, including 1kg of battery has a 25% battery fraction:

The theoretical power that is needed for propulsion is calculated as the flight speed multiplied by the drag:

For fixed-wing aircraft this formula makes sense but multi-rotors also consume propulsion power to create lift, right? The propellers create both lift AND propulsion. This is analog to an airplane which uses wings and a propulsion unit (propeller or jet) to create its lift and propulsion respectively. So just consider the propellers of a multi-rotor acting as a wing and a propulsion unit simultaneously.

We do not know the drag of the UAV and we cannot measure it. Why not? For a fixed wing aircraft you can put the aircraft in the wind tunnel and measure the drag with a force sensor with the propulsion unit producing zero thrust. Or we can measure the thrust of the propulsion unit in flight with a force cell. We can do this because the propulsion unit and the ‘lift unit’ (=wings) are separated systems. Because the propellers of a multi-rotor act both as wing and propulsion unit, we cannot simply remove them and measure the drag in a wind tunnel. This would be the same as measuring the drag of a fixed-wing aircraft without the wings. Because we cannot separate lift and thrust (= drag in regime flight), we continue with a generally accepted ratio in aviation: the maximum lift-to-drag ratio, also known as the aerodynamic efficiency or glide ratio. A glide ratio of 10 for example means, if you shut down the propulsion unit, that you can travel maximum 10 meters while descending 1 meter.

The glide ratio of a fixed-wing aircraft is easily measured this way and are in the order of 10 to 40. For helicopters, it can also be defined by gliding down in auto-rotation. Therefore the blades are given a negative pitch and they act like a wing of a glider, but then rotating. Typical ratios are around 4 or lower. Because we cannot change the pitch of the blades of multi-rotor propellers, it is not possible to determine the lift-to-drag ratio by gliding down in auto-rotation.

We’ll have to find a way to determine exact the lift-to-drag ratio for multi-rotors later on, but let’s assume for now we know the ratio and it is less than a full-size helicopter in auto-rotation (4) and more than a wingsuit (2.5).

We also know the lift, which is equal to the weight:

Rearranging former equations allows us to calculate the maximum range:

Since the propellers act as both wings and propulsion unit, it is also not possible to determine solely the efficiency of the propulsion unit. Therefore, we have to consider the two factors marked in yellow as one unknown value. We can make an educated guess however: the efficiency will always be <1 and the lift-to-drag ratio will be probably <4. Therefore, we can calculate the maximum range of a multi-rotor with 180Wh/kg batteries and 25% battery as:

The energy density of the battery and the fraction of the battery are easily measured and should be maximized to get the highest range.

but we need a way to know this number for a multi-rotor. This number is a combination of aerodynamic and electric/mechanical efficiency of a multi-rotor and would be a good benchmark to validate the capability of a multi-rotor to cover large distances.

Doing some more math with former equations allows us to determine the highest value of this number by performing some test flights at different flight speeds (yay! No wind tunnel test or force cells required, only a speed and a power measurement!)

If we want to consider only the aerodynamic performance of the multi-rotor, including the propellers, we can define a ‘theoretical glide ratio’, which we cannot achieve because auto-rotation is not possible:

With P_mech the total mechanical power delivered to the propeller shafts during cruise flight. This can be measured or calculated if the efficiency of wires, ESC and motor is known for all possible operating points.

Now the big question: what is the theoretical glide ratio of a multi-rotor? During my PhD, I performed many test flights with different multi-rotors and here are the results for a popular setup:

The best total flight efficiency of this setup was:

occurring at a total mass of 1.6 kg and a speed of 16.3 m/s. With a 25% battery fraction and 180Wh/kg batteries, this results in a range of about 20km. Because I measured the efficiency of the motor and ESC in all possible operating regimes on a test bench, I was able to also compute the theoretical glide ratio:

So that is somewhere in between a flying squirrel and a wingsuit (not too good!).

Conclusion: The range of multi-rotor can be easily computed if you know the ‘flight efficiency’, which is a combination of the efficiency and aerodynamics of the ESCs, motors, propellers and the body/arms/wings. The best flight efficiency of a conventional quadcopter was calculated from flight data as 1.2 and the accompanying ‘theoretical glide ratio’ was 2.13. People working on VTOL hybrid solutions should be able to increase this number towards glide ratios of fixed-wings (> 6). However… battery fraction is of equal importance if you want to maximize your range.

Question: what is your flight efficiency and battery fraction? I’d love to see some comparison between multi-rotors, helicopters, fixed wings and hybrids!

If you like to learn more about the performance of multi-rotors for high speeds, long range, endurance, payload capacity, you want access to all the data on different vehicles or for more specific questions contact me on bart.theys@kuleuven.be

## Comments

Ah, yes, I agree Bart. At least in the case of multirotors. I've not tested it, or really even thought about it, but I think you're probably right. A big part of the reason is that the things you do to make a multirotor hover for longer, actually hurt it's ability to fly at speed. Larger, slower propellers, with necessarily slower propeller speed. And longer arms to create drag.

In particular, the large, slow propellers are a problem. When rotary wings have lateral speed, a portion of the retreating blade ends up with zero airflow, and then even reverse airflow. So they end up actually creating downward thrust. This has to be compensated with more power being used just to achieve lift.

And the fact is, that as the blade design is optimized for better hover efficiency, typically with taper and twist, things get worse. We have all seen the really nice multirotor blades with huge paddles inboard with lots of pitch, which then narrow and get flatter as they move out. Those huge inboard twisted paddles created a lot of drag in forward flight.

Helicopters have exactly the same issue. But, typically they use narrow "slab" blades, which are less efficient in hover, but also have less of a problem in forward flight. There's a natural trade-off here. The more a rotor blade is optimized for hover, the less optimized it is for forward flight.

Racing quad designs, don't have this sort of problem. They have small, high speed propellers, and fly forward with lots of pitch, so most of the airflow remains axial through the rotor. This is why they can fly so fast. But then as we know, flight times are short.

Helicopters have another advantage to solve retreating blade loss of lift, which is that the swashplate allows for more pitch to be added to just the retreating blade, which is what is done. This does help quite a bit, until you start having really heavy reverse flow on the retreating blade in which case it is hurting. Power consumption goes up very fast beyond this point.

I'll have to think about the proposition you are making, it's interesting and I never thought about it in those terms. I have found that helicopters can fly quite far if you just keep loading them up with batteries. They don't have as much negative coefficient as multirotors do as you load them more and more.

My Procyon can fly for 40 km, probably 50. Much further than a similar size multirotor.

@ Rob, you are completely right concerning flight time. However, will multi-rotors or helicopter that are designed for maximum flight time (with low disk loadings and the corresponding control issues), also have the highest range? From my test data, it is shown that it is not the setup with the lowest disk loading that is most efficient in traveling long distances. The same holds true for full size helicopters: the glide ratio of the same helicopter improves up to a certain point when it is more heavily loaded. The flight time however, decreaes. (We are looking at CL/CD here, not (CL^1.5)/CD. Because you are the expert on UAV helicopters here, do you have some experiments on their power consumption for different flight speeds and total masses?

Bart, there is a non-obvious, but quite real correlation between multirotor flight efficiency with reduced stability.

The recipe for making a long-duration multirotor, is to use very large propellers relative to the mass. This is very obvious. Momentum theory predicts that reduced disk loading reduces hover power. But there's actually a square-root relationship here. Diminishing returns.

The other thing you need to do to increase flight time, is to maximize your battery-mass-fraction. How much of your AUW is battery. This sometimes results in frames which are structural deficient, but let's ignore that point. It also means that your motors need to be as small as possible, while still being able to supply the power needed to lift the load without going into thermal overload.

But what most overlook when dreaming about that 1-hour duration multirotor, is that as disk area goes up, the moment of inertial of the propellers goes up with roughly the 4th power of the propeller size. Meanwhile, the motor size/power does not increase. Therefore, the size-optimized motors are quite underpowered for the size of propeller they are driving. Therefore, flight stability decreases because the motors don't have enough power to accel/decel the large heavy props as is required for fixed-pitch stability.

Another aspect that is overlooked, is that as disk loading goes down, the effect of wind on the disk becomes stronger. So just as the control responsiveness as the system is going down, the strength of the disturbances is going up.

I recently read an interesting article from Jeevan about the state of the industry, and he talked about the inevitable adoption of quad-planes for increased flight duration. In the comments, Brandon Basso disagrees, and suggests that 'Mavic 3' will have 1 hour of flight time at 40mph. This is ignoring all of these real physical problems.

It's taken 3-4 years for multirotor drones, real, practical, usable ones, to increase flight time from 15 minutes to 25 minutes. Some lazily conclude that flight time will just continue to advance. It's a false assumption. What has happened is that multirotor powertrains have increased efficiency. But they have essentially run up to the brick wall of efficiency predicted by the Momentum Theory. Whereas 3-4 years ago, they were achieving a Figure of Merit of about 40%, they are now reaching about 70%. This is why flight times have increased. But this can only go so far. Full size helicopters, with millions of dollars invested in aerodynamics, and benefiting from some important scale factor advantages re: Reynolds Number only achieve about 90% efficiency. So it would be hard to imagine that multirotor efficiency will get better than that. Therefore, there is no chance that multirotor flight time performance will improve 3-fold over where it is now.

There is of course some opportunities in energy storage density. Battery Tech. Lots of technology teased here over the years, but nothing on the market yet.

There's also some improvements possible with mass-fraction. Basically reductions in payload weight and airframe weight. And the Mavic actually took a huge leap here. DJI have realized that the single most important aspect of multirotor design, is that the entire system hinges on the gimbal design. The entire system scales with the cube of the stabilized imager size. The smaller your camera head is, the smaller your gimbal structure and motors can be. And the smaller those are, the smaller the rest of it can be. This is why systems that rely on a hand-held camera on the end of a gimbal, like the Karma, are at a MASSIVE disadvantage.

But the structure itself, does not have as much room for improvement. There's a limit how flimsy you can make these things before they become too fragile. Particularly for consumer or commercial usage.

@ Auturgy: What is the link with stability that you are thinking of? Also, I agree that the real glide ratio, as in: how far can you glide with power of, is <1 but what I show in the article is the theoretical L/D ratio. It is a measure for the amount of power required to move a mass with a certain speed through the air. But perhaps I that was not so clear in the article?

@Bart - just a comment relevant to your Total Flight Efficiency. Don't use it as a design input for multirotors. The resultant aircraft would be unstable in even light winds. Also, your assumption that L/D for a multirotor would be better than a wingsuit is probably very inaccurate. For autorotation to occur not only do you need to drop collective, but you need to be able to freewheel. Multicopters often can't, so you lose almost all energy from the props in a very short time period. No energy = no spin = no lift = L/D < 1

Thx for the comments,

@ Frederic: the airplane from your 'back of the envelope' calculations has a (what I call) 'total flight efficiency' of 8.3. So it would be able to fly about 4 times further than the F450 quadcopter with the same % of batteries. That is feasible. Does anyone has actual data on: mass, speed, power of other UAVs?

@Gary:I believe you are right but I'd like to see the calculation for the total flight efficiency and the glide ratio for one or more helicopters from actual flight data before we can make this conclusion.

Hi Bart,

Cutting straight to chase from experiences so far.

Highest efficiency and longest range to fixed wing by considerable margin - big surprise - not.

Helicopter is generally considerably more efficient than (and longer range than multirotor).

Simple fact is that bigger single rotor is generally way more efficient than many smaller rotors.

And multicopters do not control well with large (non-variable pitch or cyclic) rotors (over 24") in any case.

Heli has disadvantage of requiring real time pitch and cyclic control vbut compensates with increased efficiency and flight capability.

(See Rob Lefebvres Procyon).

https://novaerial.com/

Direct comparisons between fixed wing and rotor craft are hard to model, because fixed wing can get down to no power at all under the right conditions and as you stated, rotorcraft require power to provide lift at least some of the time, although good helis can get this requirement down to a relatively small amount when flying at the most favorable velocity.

Although in one sense rotors are propellers, prioviding direct thrust, heli rotors als behave very much as continuously adapting wings where the lift can be provided by their horizontal velocity through the air rather than directly consumed as motor power.

Think gyrocopters.

As for flying squirrels, they are arguably not flying at all, rather parachuting with a fairly controllable parachute (skin flaps) and as such are completely un-powered during flight and descending only to calculate their efficiency you would really need to take into account calories (watts) used climbing the tree in the first place.

Hang glider pilots too.

Best regards,

Gary

@Gary. is that the "wing" project you are talking about? I read some rumor but from people directly working there it is business as usual...

interesting study!

I have approached the efficiency problem from the fixed wing point of view.

first I came to the conclusion that the range is not the factor to optimize I prefer the range x speed. you want to fly far and fast. optimizing is always a trade off so optimizing the product is the way to go I think.

with a fixed wing, if you optimize the range only you cannot beat a design like the deadalus ( the human powered plane that crossed part of the mediteranea) it needs 250W to fly and you can replace the 70Kg of pilot with 70Kg of battery .it would fly more than 1000Km but would need more than 42H for that!

then I am not sure that we should use "glide ratio" terminology in the context of a multi copter because it creates a confusing mental image. what you describe is an ideal lift to drag ratio that exist only when the motor is on, kind of strange for a "glider" ;-)

if we compare with a fixed wing of equivalent sizing ( 1.6Kg, 25% battery weight, 16m/s) with an average non optimized design ( 2m, AR=10, e=0.7, wing area=50% total wetted area, propulsion efficiency=0.6...) the back of the envelope calculation gives power for level flight ~30W so a range of ~140km. it is in no way a precise model from actual data like you, but it gives an order of magnitude

cheers

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