Theoretical power for flight
I want to look at the potential energy a glider loses in still air as a way to think about the power needed to power level flight. Imagine we make a glider and add weight so it has our total expected airplane mass. Then we can measure the speed and glide slope and estimate how much power it will need for powered flight.
The potential energy is m*g*h. Mass times gravity times height.
power = energy/time
For a glider the interesting information is:
m = mass of the glider
v = velocity = airspeed
l/d = lift divided by drag = glide ratio = forward distance / down distance
GlideRatio = AirSpeed/SinkRate
g = 9.8 meters/second^2
SinkRate = v / GlideRatio = how fast glider is loosing altitude
TheoreticalPower = m*g * SinkRate
watt = joule/second = 1 Kg * meter^2 / second^3
Lets do a concrete example with:
m = 1 Kg
v = 7 meters/second (about 15 mph)
l/d = 10
TheoreticalPower = 1 Kg * 9.8 meters/second^2 * 7 meters/second / 10
= 6.86 watts
If we had perfect propulsion it should take about 6.86 watts to keep such a plane flying level. We need to look at the propulsion efficiency to get real world numbers. This is the product of the efficiency of the motor and the efficiency of the propeller.
PropulsionEfficiency = MotorEfficiency * PropellerEfficiency
Lets use these numbers for now:
MotorEfficiency = 85% = 0.85
PropellerEfficiency = 80% = 0.8
PropulsionEfficiency = 0.85 * 0.8 = 0.68 = 68%
Formula for total real world power:
RealWorldPower = TheoreticalPower / PropulsionEfficiency
So in our example:
RealWorldPower = 6.86/0.68 = 10 watts
It is clear from the above math that the power needed scales linearly with the mass. If we went from a 1 Kg design to 2 Kg design we would need twice the watts, assuming all else was the same.
It is also clear that if we designed for twice the speed we would need need twice the power, all else being equal.
If we get twice the L/D we can use half the power. So an efficient wing is an important factor.
Note that adding more weight to a particular glider design will make that glider fly faster and have more weight. In this paper they state, "They both increase with the square root out of the weight. So if the
weight would double both the True airspeed and vertical speed increase
by the square root out of two which is about 1,41. " So simply adding weight to an existing design results in a worse than linear power increase.
Anyway, if we want to run on solar power, designing for light and slow is ideal.
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