In this instructable some parameter that decide flying time is discussed.

## Step 1: Propeller Size

**Total area for propelling:**To make X kg of object fly we need push X kg of air downward. To keep object in air velocity of pushed air do not matter but the thing matter is how much air is getting pushed (mass flow rate) downward.*[Ideally]*

for an example two quad copter are shown in figure. Assume both has same weight (of 1.2 kg) one has all propeller of 10 Inch and another has all propeller of 7 Inch. total area covered by 10 inch prop is approx 2000 cm^2 (let name it quad-2000) and area covered by 7 inch prop is 1000 cm^2 (quad-1000).

(cc=cubic centimeter =cm^3)

So,

To keep quad copter in air 1.2 Kg of air need to be pushed downward per seconds which is equivalent to 1 cubic meter (or 1000000 cc of air).

for a qaud-2000 , 10000000cc of air need to pass from 2000 cm^2 of area per second.so it can be directly calculate:

=(10,00,000 cm^3 /s) / (2000 cm^2)

=500 cm/s

=5 m/s

same way for quad having 1000 cc air speed required =10 m/s

power requirement for quad to be airborne:

energy imparted to air per second(quad-2000)= 0.5 (density*cross sectional area* velocity^3)

=15

energy imparted to air per second(quad-1000)= 60

quad -1000 need ideally four times more power to fly with compared to quad-2000.

so **bigger the prop higher flying time is achievable.**

[these power rating are for ideal calculation which are far smaller than real values]

BUT,

- a qaud with bigger prop would have lower maneuverability as inertia of prop is higher.
- higher prop size leads higher weight , and bigger prop runs at relatively lower speed(lower Kv) need bigger size of motors so this increment of weight possibly reduce flight time after some limit.

## Step 2: Understand Effect of Weight and Choosing Battery Weight:

It is very straight forward and intuitive that any flying machine having lower weight would fly for longer duration.

Quantitatively flying time is inversely proportional with 1.5 power of weight. so if increase weight of quad 4 time (keeping battery spec same) would reduce flying time by factor of 4^1.5= 8 times.

**Choosing battery size:**

- if we increase size (weight) of battery we have more energy to use on board so this increase flying time. But weight also increases which reduces flying time.

so it is required to understand sweet point for this battery and frame weight ratios.

in next step mathematical derivation is explained

if you interested only in result and not in boring math skip next step.

## Step 3: Mathematical Derivation to Pick Best Battery Pack:

Above image represents derivation for best battery weight.

here a parameter to consider is weight ratio which is equal to ratio of battery weight to frame weight (without battery).

in Eq(1) weight and thrust is compared. than in Eq(2) energy in battery is compared with energy imparted into flow of air. in eq(3) both eqaution is combined. in eq (5) flight time dependence on battery and frame mass is derived.

In eq(7) optimized value for battery and frame weight is shown.

## Step 4: Conclusion for Battery Pack Selecion:

**weight ratio =battery weight / frame weight;**

(frame weight =weight of quad copter without battery)

Point for Maximum flight time is when weight of battery is double (weight ratio= 2) than of quad copter frame. Ideally if this condition satisfied maximum flight time can be achieved.

But from above plot it is clear that even at weight ratio 1 around 93% (of maximum possible) flight time can be achieved. So it not worth to increase battery weight same as of frame weight to just to increase 7% of flight time (cost on battery would be almost doubled).

**(Increasing weight ratio = more money)**

It can be seen that after weight ratio of 0.6-0.7 (60%-70% weight) no noticeable increment occurs. So spending extra money on battery after weight ratio 0.7 is not worthy.

## Step 5: Examples:

in above image example of various commercial quad copter is shown.

except above parameter flying time is also depends on quality of electronics and prop balancing.

*above data is taken from different web sites for different quad copter so exact value may be slightly differ from mentioned data.

## 5 Discussions

1 year ago

The math is not quite correct, it assumes air is moved at constant speed, while we accelerate particles from zero to v. Actually thrust is proportional to (Area*Power)^2/3 and flight time ~1/sqrt(Mass). Mbat=2Mcraft is correct

1 year ago

Lift is proportional to the 'wing' (prop) area, air density resulting from altitude and temperature, the blades' speed and a coefficient dependent on the 'wing's shape and a Lift Coefficient. The Lift formula is L = (S * V squared *Rho * CL) / 2 where

L = Lift

S = Area

V = Speed

Rho = Air density

CL = Lift Coefficient.

Controlling the quad is done by changing the prop's speed which translates into Lift change and movement due to vectored Lifts.

1 year ago

The prop is a rotating wing hence it lifts the quad due to airflow generated above and below the prop's surfaces and the resultant pressure difference.

Seems that your are rewriting aerodynamics...

2 years ago

Cool!

i love seeing all the math behind choosing electronics

this'll help me figure out frame weight for the tricopter i'm building

Reply 2 years ago

Thanks lot