Fix my propeller thrust equation derivation

[ElectricRCAircraftGuy]

My mass flow rate equation is wrong. This one: https://www.electricrcaircraftguy.com/2014/04/propeller-static-dynamic-thrust-equation-background.html

Someone let me know that:

equation 5 for mass flow rate, m_dot = ρ*A*Ve, is incorrect. The velocity of the air at the rotor disk, (let's call it: Vdisk), is halfway between Vac and Ve. In other words, Vdisk = Ve - (1/2)*ΔV, (or Vdisk = Vac + (1/2)*ΔV). Thus, the correct equation according to conservation of mass, should be: m_dot = ρ*A*Vdisk = ρ*Aac*Vac = ρ*Ae*Ve.

It can be shown that the correct equation for the thrust, in terms of the inlet and outlet velocities Vac and Ve, is: F = (1/2)*ρ*A*(Ve^2 - Vac^2). See the following NASA and MIT pages for the correct derivations:

https://www.grc.nasa.gov/www/k-12/airplane/propth.html

https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node86.html#SECTION06373000000000000000

Other useful links:

1. https://talkrational.org/tikal/viewtopic.php?t=36
2. https://talkrational.org/index.php/topic,24.msg468072.html#msg468072

Risking My Life To Settle A Physics Debate https://www.youtube.com/watch?v=jyQwgBAaBag

I should read this too, probably before the MIT article above even:

See: https://en.wikipedia.org/wiki/Betz%27s_law

[ElectricRCAircraftGuy]

Links I should look at:

A private email with some notes (link only works for me): https://mail.google.com/mail/u/0/#inbox/FMfcgzGsnBlCLGmxMQGGGDpRRzSKcNGg

1. [x] ***** Propeller Thrust: https://www.grc.nasa.gov/www/k-12/airplane/propth.html - derives it using total pressure, where Ptot = P + q, and q is dynamic pressure, or 1/2*rho*v^2, as I know from my aeronautical engineering undergrad at USAFA.
2. [ ] ***** General thrust equation: https://www.grc.nasa.gov/www/k-12/BGP/thrsteq.html <===
3. [ ] ***** "11.7 Performance of Propellers" and "11.7.3 Actuator Disk Theory" - https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node86.html#SECTION06373000000000000000 <===
4. [ ] ***** "In aerodynamics, Betz's law indicates the maximum power that can be extracted from the wind": https://en.wikipedia.org/wiki/Betz%27s_law - should be a nice accompaniment to the MIT article above. <===
5. [x] https://www.grc.nasa.gov/www/k-12/BGP/mass.html
6. [x] https://www.grc.nasa.gov/www/k-12/BGP/mflow.html
7. [ ] compressible flows: https://www.grc.nasa.gov/www/k-12/BGP/mflchk.html
8. [ ] https://www.grc.nasa.gov/www/k-12/BGP/bgp.html
9. Google search for "equation for mass flow rate"
10. [x] https://www.youtube.com/watch?v=qebMrMt4240 - spool paradox - useful to understand counter-intuitive things and consider torque and pivot points; this may play into my understanding of the cart's wheels driving the propeller with wind as the power, while traveling faster than wind
11. https://talkrational.org/tikal/viewtopic.php?t=36&sid=80012ab028f85d4a48686c6893c70591&start=125
12. https://talkrational.org/index.php/topic,24.msg468072.html#msg468072 - mentions my name, "Gabriel Staples", a few times.
    1. See posts by "Brother Daniel"

DDWFTTW = "directly downwind faster than the wind"

I have much work to do on this. I'd like to analyze the Blackbird land yacht DDWFTTW too, which I first saw on Veritasium:

1. https://en.wikipedia.org/wiki/Blackbird_(wind-powered_vehicle)
2. https://www.youtube.com/watch?v=jyQwgBAaBag
3. https://www.youtube.com/watch?v=CkhBskovitQ
4. Google search for "DDWFTTW"
   1. https://www.wired.com/2010/08/ddwfttw/ - "A Long, Strange Trip Downwind Faster Than the Wind", by Rick Cavallard

Other vehicles/physics concepts to consider include sailing and kiteboarding. In both cases, you can go faster than the wind. When kiteboarding, you can also actively "fly" the airfoil in a low figure 8 pattern right above the water in the downwind direction to "power up" the kite, making it produce a ton more thrust than just sitting stationary in the downwind position. This is because when stationary, it acts under the principle of drag, where F_max is based on the speed of the wind alone. But, when actively flying the kite in a figure 8, it also has a ton of forward velocity, causing it to produce force based on the principle of drag (wind hitting it) as well as airfoil lift due to forward motion. I think there are probably a lot of similarities between the propeller on the Blackbird and a kiteboarding foil.

For the kite (but not the propeller), you are, perhaps, using both gravity and wind: the gravity accelerates the kite as it dives down, and the the wind pulls the kite back up (like a regular park kite). Maybe... I should also consider the Google X Makani kite, which was intended to fly a canted circle type pattern (I think), similar to a kiteboard foil, in order to produce electricity.

[ElectricRCAircraftGuy]

Keep in mind that even though my propeller thrust derivation is wrong, my empirical corrections have clearly corrected it well enough, as it works.

From here: https://www.electricrcaircraftguy.com/2014/04/propeller-static-dynamic-thrust-equation-background.html

See:

Static thrust: my equation vs test results:

dynamic thrust: my equation vs wind tunnel tests:

This is also correct, though poorly labelled with the arrows on the tiny diagram:

[ElectricRCAircraftGuy]

Work to do

1. [ ] study propeller thrust equations above by MIT, NASA, and Betz's Law on Wikipedia.
2. [ ] fix my propeller thrust equation derivation
3. [ ] open source my thrust data, spreadsheet, etc.
4. [ ] redo my empirical correction factors
5. [ ] write software and tools to perform blade element theory on propellers
6. [ ] apply physics modeling to the Blackbird land yacht
7. [ ] write articles and stuff as I go

[ElectricRCAircraftGuy]

Other things to consider:

1. A sailboat tacking at a large angle (ex: 90 degrees) to the wind is like a really shallow leverage ramp, where a very strong force over a short height pushes the sailboat very fast a very long distance.
2. This spool paradox is basically leverage ramps: https://www.youtube.com/watch?v=qebMrMt4240
3. So is the propeller wind car I think.

Also:

1. Just as the blade's incident AoA (angle of attack) approaches zero degrees as an airplane's forward velocity increases, blowing wind behind the propeller towards the propeller (like wind blowing towards the land yacht), does the opposite: increasing the angle of attack, and thereby increasing required torque, and thrust output, as the wind behind speed increases. This helps explain the phenomenon of "how does the land yacht speed up as tail wind speed increases?" Well: the AoA is increased as tail wind increases, giving the prop more "bite" into the air.
   1. It would be really insightful to plot the velocity vectors (wind (V_a/c shown in my diagram above) + perpendicular propeller blade velocity due to spinning (Vr in my diagram above)), plus their sum (V_infinity in the diagram above), as the vehicle goes from 0x the wind speed through 1x and then up to 3x the wind speed. <== key takeaway! So, do this. Graphically show it as the vehicle speeds up. Use my (ideally, fixed--TODO) propeller thrust equation to help estimate thrust and come up with real numbers throughout the process.

[ElectricRCAircraftGuy]

See also here: https://www.tytorobotics.com/blogs/articles/how-to-calculate-propeller-thrust

They mention my "Dynamic Propeller Thrust Equation" by "Electric Aircraft Guy". Their static propeller thrust equation is from NASA, and contains the corrections I think my equation derivation needs to be more correct, namely: the 0.5 multiplier part.

[ElectricRCAircraftGuy]

1. [ ] read & study this too: https://web.mit.edu/drela/Public/web/qprop/motorprop.pdf