- #1

- 5

- 0

## Main Question or Discussion Point

I'm having some issues finding the relationship between power and thrust of a motor of a standard propeller based aeroplae.

I've found two simple relationships that both make sense, but both (seem to) contradict each other.

Thrust is equivelant to change in momentum:

T = Mdot*v

Power is the integral of thrust with respect to velocity:

P = int(Mdot*v)dv = 0.5*Mdot*v^2

Mass flow rate is ρAv, hence:

T = ρAv^2 and

P = 0.5*ρAv^3

Or, rearranging gives:

P^2 = (T^3)/(4ρA)

I can follow that fine and it makes sense to me. But there's also an alternative solution:

Work = Force * distance, so:

Power = Work/Time = Force *Velocity = Tv = ρAv^3

Rearranging gives:

P^2 = (T^3)/(ρA)

This is the same as before but without the factor of 1/4. Both make sense to me but obviously can't both be correct. Could anyone provide some guidance on the matter?

(If it helps, both formulas are contained on wikipedia: http://en.wikipedia.org/wiki/Thrust . My scenario concerns a motor spinning a propeller to provide thrust for an aeroplane, which seems to be more in line with the first equation. But given that the first equation gives a lower power requirement I'm really more inclined to go with the second just to be safe.)

I've found two simple relationships that both make sense, but both (seem to) contradict each other.

Thrust is equivelant to change in momentum:

T = Mdot*v

Power is the integral of thrust with respect to velocity:

P = int(Mdot*v)dv = 0.5*Mdot*v^2

Mass flow rate is ρAv, hence:

T = ρAv^2 and

P = 0.5*ρAv^3

Or, rearranging gives:

P^2 = (T^3)/(4ρA)

I can follow that fine and it makes sense to me. But there's also an alternative solution:

Work = Force * distance, so:

Power = Work/Time = Force *Velocity = Tv = ρAv^3

Rearranging gives:

P^2 = (T^3)/(ρA)

This is the same as before but without the factor of 1/4. Both make sense to me but obviously can't both be correct. Could anyone provide some guidance on the matter?

(If it helps, both formulas are contained on wikipedia: http://en.wikipedia.org/wiki/Thrust . My scenario concerns a motor spinning a propeller to provide thrust for an aeroplane, which seems to be more in line with the first equation. But given that the first equation gives a lower power requirement I'm really more inclined to go with the second just to be safe.)