Calculating Air Velocity in an Elastic Collision

[jaguar7]

F=force, m=mass, a=acceleration, V=Volume, D=Density, dv/dt=(change in velocity)/(change in time)

F=ma

D=m/V
m=VD

a = dv/dt

F=VDdv/dt

= Volume * Density * (change in velocity) / (change in time)

= (Volume of Air Moved) * (Density of the Air) * (the final velocity of the air) / (change in time)

(final velocity of the air) = ? (How do you determine this? It's elastic collision I'm guessing?? But I don't know what to do...)

 

[jaguar7]

Update:

If we can outline a path that the propeller travels over time, then we can calculate weight/time of the fluid it displaces.

Weight is a force? Or do we need mass? If a propellor displaces a certain amount of volume of a fluid substance, how do we calculate force, if we know the density of the substance? We don't know the distance it was displaced. We just know that it was displaced. We can calculate that a certain mass was displaced over a certain amount of time. I feel like this should be equatable to force somehow. But I guess if I think about rocket fuel being displaced from a rocket shiip, we would still need to know the final velocity of the rocket fuel in order to calculate the force.

If we know the velocity of the propeller (obviously it's changing along the radius), then can we set up any kind of equation? i.e. "The propeller travels at speed, V, imparting the surrounding substance with momentum, p"?

Also, does the presence of a rotating propeller in the substance change the pressure of the surrounding substance?

 

[EWH]

Do you have any numbers to go with this problem, or are you just supposed to set up the equation?
http://en.wikipedia.org/wiki/Betz'_law has some relevant derivations (in reverse - it's about turbines, not propellers.)

Edit: here's an equation found by googling "propeller equations":
T= π/4*D²*(v+Δv/2)*ρ*Δv
T thrust [N]
D propeller diameter [m]
v velocity of incoming flow [m/s]
Δv additional velocity, acceleration by propeller [m/s]
ρ density of fluid [kg/m³] (air: ≈ 1.225 kg/m³, less at altitude or when hot)

see http://www.mh-aerotools.de/airfoils/propuls4.htm for more on power, etc.

 

[jaguar7]

That equation looks oversimplified to me, as it doesn't include angle of attack or area of the airfoil, or anything like that. I had found a seemingly oversimplistic equation before, but at this point, I would like to see a derivation.

Additional Information:

"Pressure arises because each molecule that bounces off the surface transfers momentum to the body. If a particle of mass, m, hits the body "straight-on" and bounces off, it transfers momentum of the amount 2mc where c is the speed of the molecule. The pressure is then proportional to the number of molecules striking a unit area of the surface per unit time, (Number density*c), times the momentum transfer per particle, (~mc) or: p = k1 ρ c2

Note that since temperature is defined as proportional to the mean kinetic energy of the molecules, T = k2 c2. So we expect: p = k ρ T, the perfect gas relation.

The component of molecular velocity normal to the surface is what is really needed in the above expression, and if the body is moving, we must add its velocity to the molecular velocity measured in the "fluid-fixed" reference frame."

- http://www.desktop.aero/appliedaero/fundamentals/pressures.html

-- From this, I should be able to continue working from where I left off [that is, F = (Volume of Air Moved) * (Density of the Air) * (the final velocity of the air) / (change in time)].

Also, I learned that pressure is actually Force per Area. So that seems like it might be useful if I could somehow find the change in pressure from the front to the back of the propeller.

 

[EWH]

All that stuff is factored into the initial velocity and the change in velocity.
Here is a plausible derivation (with me cheating by knowing the expected answer):
Prove:
T= π/4*D^2*(v+Δv/2)*ρ*Δv
(substitutions indented)
F=ma
m= ρ*V
V/Δt=v*A
| V=v*A*Δt
| m=ρ*v*A*Δt
A=π/4*D^2
| m=ρ*v*π/4*D^2*Δt
| F=ρ*v*π/4*D^2*Δt*a
a=Δv/Δt
| F=ρ*v*π/4*D^2*Δt*Δv/Δt
| F=ρ*v*π/4*D^2*Δv
v_avg = v_0 +1/2*a*Δt (constant acceleration)
| = v_0 +1/2*Δv/Δt*Δt
| = v_0 + Δv/2
v= v_0
| v_avg = v + Δv/2
| F=ρ*(v + Δv/2)*π/4*D^2*Δv
F=T
So
| T=π/4*D^2*(v+Δv/2)*ρ*Δv
as stated in the original equation.

You are right to be a little suspicious, the thrust is easier to measure directly than the v and Δv, especially with the v_f (after the prop) being turbulent and even the average v_f varying radially

 

[jaguar7]

okay,

How do you find delta v (change in velocity)? Given propeller shape, pitch angle, diameter, area, etc. and air pressure? That was actually my original question. For some reason I didn't notice that ⌂v of the air was in the equation you gave.

Also, I'm reading the link you gave, and I want to clarify that one variable that is unknown is the final velocity of the craft at "full speed."

Also, could you name your variables? Specifically, I'm guessing capital V is volume? But then you say V/⌂t=v*A ... I'm just not sure about some of the variables.

 

[EWH]

Well, I think there won't be any closed-form solution for Δv or thrust involving the prop shape and pitch angle. It's just too messy - real aerodynamics. Maybe if those get condensed into a coefficient of lift, and you have RPM, but still messy. Is this an assigned problem or just something you're thinking about?

In the real world you either do complicated CFD simulations, or use rules of thumb (perhaps from software - try http://www.drivecalc.de/PropCalc/PCHelp/Help.html ), or you use something off-the-shelf that has test data and specifications, or you just try things out with prototypes and measure the thrust. Usually you'll be limited in diameter that will fit on the plane, the engine power and RPM, and you'll know about what speed you will be going (e.g. static thrust). These, together with typical prop. design performance factors give a rule of thumb estimate of thrust.

If you have the ambient pressure and the pressure behind the prop, of course it's easy, just pressure differential times swept area.

***
F force, N
m mass, kg
a acceleration, m/s^2
ρ rho, density, kg/m^3
V volume, m^3
Δt time, a short interval, should really be dt
v velocity, m/s
A propeller swept area, m^2
π pi
D prop. diameter, m
Δv change in velocity, should really be dv
v_avg average v over the period that the prop. is accelerating the air
v_0 initial v of the incoming air
T thrust, N
v_f in my last post meaning the v of the air after it has been accelerated, not the final velocity of the aircraft.

Prove:
T= π/4*D^2*(v+Δv/2)*ρ*Δv
(substitutions indented)
F=ma (Newton)
m= ρ*V
V/Δt=v*A (volume per unit time equals the velocity of the incoming stream times the area)
| V=v*A*Δt
| m=ρ*v*A*Δt
A=π/4*D^2 (circular area expressed as diameter rather than radius)
| m=ρ*v*π/4*D^2*Δt
| F=ρ*v*π/4*D^2*Δt*a
a=Δv/Δt (definition of acceleration)
| F=ρ*v*π/4*D^2*Δt*Δv/Δt
| F=ρ*v*π/4*D^2*Δv
v_avg = v_0 +1/2*a*Δt (constant acceleration formula from every standard textbook, using Δt because of the short acceleration period and to get the Δt canceled out of the final equation while keeping the (v+Δv/2) term. Yes, it's sort of cheating a little.)
| = v_0 +1/2*Δv/Δt*Δt
| = v_0 + Δv/2
v= v_0 (more cheating to get the answer I want)
| v_avg = v + Δv/2
| F=ρ*(v + Δv/2)*π/4*D^2*Δv
F=T
So
| T=π/4*D^2*(v+Δv/2)*ρ*Δv
as stated in the original equation.

Edit: Here's a reference http://www.scribd.com/doc/68114590/Journal-04-MDOpropulsion to show you just how complicated designing real propellers is.