The Great Copter’s Theorem usage examples

The Great Copter’s Theorem© (GCT) shows maximal possible time of flight. In real implementations it is lower because of 1) at the near maximal flight times, hovering time slowly depends on energy stored in the battery or fuel, so it is not very efficient to have huge battery,  better compromise would be 1/3 of maximal weight for the price of slight drop (80%) of hovering time 2) Propeller and motor efficiencies (Figure Of Merit, or FOM) are lower than theoretically possible.

Here is online tool  to see hovering time vs battery capacitance or fuel tank.

Nonetheless GCT can predict real flight time if the following corrections are made:

  1. We take battery or fuel energy of ~1/3 of maximal (maximal time will drop to 80%)
  2. Efficiency of motor and propeller is 0.3-0.5.

Here is how it works:

Your plane crashed on an island. You found a helicopter, the label says “Robinson R66, empty weight 600 kg, rotor diameter 10 m”. You would like to know if you can reach land (how long you can fly).

Using GCT, we know that fuel weight to get maximal time is 600*2=1200 kg. Maximal time is w(Wh/kg)*10(m)*sqrt(1/600kg)~5000*0.4~2000 minutes. (do not expect units rules, see GCT page), (for estimation of w for gasoline see Aero-X chapter below).

Apply above rules 1): realistic fuel weight is 1200kg/3=400kg (140 gallons). Time will be less in approximately 0.8 (rule 1).  2000 min*0.8=1600 min. Apply rule 2) 1600 min*0.3=480 min=8 hours.

So you can expect that flight takes 140gal/8hours=17 gal/hour. Specifications for Robinson R66 shows 20 gals/hour. Our simple calculations give reasonable values, also we took empty weight and used hovering (not real flight time). And we used only 2 basic and measurable parameters of the copter!

this example plot

If electric copter is found, similar estimation can be done, because GCT is independent on fuel type, all you need is keep in mind energy density of different types of fuel (or batteries)!

Below are more detailed results for some electric, gasoline or man powered vehicles.

 

Tiny Whoop (added  on 03/01/2018)

Here is most recent estimation for 0716 brushed tiny whoop

this example plot

DJI Mavic Pro.

image001

Mcopter=0.49 kg; w=182 Wh/kg; D=0.211 m; Nrotor =4

Overall FOM (real hovering time -to theoretically possible time) is 35%. This value is typical for copters of this type and is determined by loses of electricity-to-mechanical power conversion and air stream of propellers (side flow). Under no circumstances, it is not possible to get hovering time over 38 min with battery mods (max possible time is reached when 160Wh battery is used (weight is 1kg)).

this example plot

Volocopter (electric)

valcopter

Mcopter=450 kg; w=175 Wh/kg; D=2.5 m; Nrotor =18  (27min)

Overall FOM  is 50%.

It is not possible to get hovering time over 27 min (with LiIon battery). Even if huge 400 kg battery is used.

this example plot  assuming battery 15s 800Ah~42 kWh

 

Ehang 184 (electric)

ehung

Mcopter=340 kg; w=180 Wh/kg (assumption); D=1.6 m; Nrotor =8

FOM (real to theoretically possible) is 70% (25 min declared time). Published data do not contain type of the battery, we assumed that this is good LiIon. FOM seems too high, so either they use another type of the battery or published hovering time is a fake (I think so, because I did not find any video with tests under load). Under no circumstances, it is not possible to get hovering time over 38 min (even if efficiency is so unbelievably high). I would expect around 20 min of real time.

  this example plot  assuming battery 15s 800Ah~42 kWh

 

Scorpion 3 (electric)

scorpion

Mcopter=200kg (best estimation); w=180 Wh/kg (assumption); D=1.0 m; Nrotor =4

Not much data can be found. Company says 30 minutes and 140 kg of load. Even if we assume weight of all motors and the frame is 60 kg and 100% of FOM efficiency (remember, typical value is 30%) we’ll get 27 minutes even if we will use huge 400 kg  battery.

Looking at the published picture of the hoverbike, more realistic numbers are FOM<50%  10 kWh battery. It gives us estimation of 4-5 minutes of flight time.

A few minutes is a top for electric hoverbikes. Not even worse a try! (Unless other type of fuel is used, see next chapter).

this example plot assuming  battery 15s 190Ah~10 kWh

 

OK, let’s have a look at another type of copters. Gasoline powered and man-powered.

Aero-X (gasoline)

airx

claimed time of 75 minutes can be achieved with 6-15 gals

Upper plot is zoomed version of the bottom plot to show realistic tank volumes.

Gasoline has much better energy density compared to batteries (x30). To compare gasoline and electrical engines I used the following: All electric Nissan Leaf takes 34 kWh per 100 miles. Similar good gasoline vehicle has 50 mpg, so 1 gallon is equivalent to 17 kWh (rail-to-rail, including motors efficiencies). Energy density is 17kWh/2.64kg=5300 Wh/kg.

Aero-X parameters:

Mcopter=356kg (vehicle) + 140 kg (load)=490 kg; w=5300 Wh/kg; D=2 m; Nrotor =2

With these parameters, we can expect 6 hour of maximal flight time! To achieve that we will need a tank of 300 gals (830 l). Company claims 75 minutes, this corresponds to the tank capacity of 22kg (8 gals) (100% efficiency), using our typical value of 50% we get 48kg (17 gals). This is possible! May be even better, because of near-the-ground effect.

this example plot

Atlas (man powered)

image014

atlas_graphs

Sure, we cannot have set of drivers, who can provide same energy density and have continuous distribution of their weights. For our purposes to plot the graph similar to the above and estimate maximum possible hovering time we reasonably can suggest that there is a hypothetical driver whose ability to produce energy linearly depends on his weight. We know that good sportsmen of 80 kg can produce 1000 W of power for 1 minutes (https://youtu.be/S4O5voOCqAQ). So, we say we have a driver with energy density of w=1000W*(1/60)/80kg=0.2 Wh/kg

Mcopter= 55kg; w=0.2 Wh/kg; D=10.2 m; Nrotor =4

Our estimation shows that for the parameters of the Atlas copter we are at maximum of possible hovering time. There is no chance to fly it longer with more powerful (but heavier) sportsmen. Also we are close to the results obtained in the experiment: 60 seconds of hovering time. This may get possible if we consider proximity to the ground, it gives 2-3 times enhancement.

 

this example plot

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