spork":2m1ktvzp said:
One of the main problems I'm having with the derivation is that you're mixing units. The thrust from the propeller will be a simple force. The "force" required to turn the prop will be a force at a specified distance (e.g. 10 ft-lbs).
True, I did do a little hand-waving here to keep it simple. Look at it from this perspective:
Wheels have "wheel-power" equal to 100% wheel-power units (what this unit is in the real world doesn't matter)
The generator can convert 100% "wheel-power" into 100% "gen-power" units (again, what this unit is doesn't matter) The prop can convert 100% "gen-power" to 100% thrust (once again, the unit is immaterial)
I was able to ignore the units because in the ideal case it is the relationship between these things that is important. In other words, what "wheel-power" power is doesn't matter, since the relationship between that and "gen-power" and thrust will be linear. If 100% "wheel-power" created X thrust, then .5X thrust will take 50% "wheel-power". All my "math" is in thrust units (who cares what the units are since they cancel out in the comparison anyway).
I admit it is a sloppy way to do science, but this is a back-of-the-envelope calculation used to demonstrate a point, not give a numerical answer. But thanks for calling me on it, I should have listed my assumptions out up-front.
spork":2m1ktvzp said:
In this case "m" has to be "m-dot" or mass flow rate (i.e. mass/time).
Once again, my bad for a little hand-waiving. In both the cases I assumed "m" would not change. I was specifically trying to make everything a constant except the velocities. Is this a bad assumption? (I suspect it may be)
spork":2m1ktvzp said:
Again, F1 is force and power is force x velocity.
Once again it doesn't matter what F's units are, since I am comparing F to F. As long as I do not try to directly compare F to any other values other than F, I should be okay dropping units.
Assuming an ideal prop, the thrust created by the prop will be equivalent to the power used to turn the prop. No loss.
Sorry, I keep saying "equal to" and "equivalent" when I actually mean "proportional to". The bottom line is, if the prop is 100% efficient there should be no loss, 100%=100%.
We can't look at this as the prop "blocking" the tailwind. The prop is immersed in the fluid and operating in the normal way a prop does in air.
I was thinking about this last night, because I wasn't satisfied with my own reasoning. The tailwind must have a effect on the props ability to create thrust. There are only 2 forces (simplified) acting on the cart. A retarding force, and an accelerating force. The retarding force consists drag, internal friction in the system, and the power being taken by the generator to turn the prop (BTW, I am using "generator" as shorthand for "whatever is transferring power from the wheels to the prop be it mechanical or other"). The accelerating force can come from wind blowing directly on the cart, and thrust from the prop.
I am going to drop direct wind, since this is only going to be a player a <wind speed. This model specifically has to work above winds speed, so direct wind shouldn't play a part. Also I am going to drop internal friction & drag, because our system is ideal.
So, now our only forces acting on the system are power taken by the generator, and thrust created by the prop. Power taken by the generator must be constant at a given speed, so I can't change that. Therefore, somehow the tailwind is increasing thrust created by the prop. Which brings me back to the thrust equation.
spork":2m1ktvzp said:
With a tailwind, the prop is operating closer to the static case (think of a plane sitting still on the runway and revving its engine). This gives a greater AOA (angle of attack) of the blades on the air vs. blades trying to get a bite into air that's already flowing rapidly through the prop disk. Thus greater thrust.
The second way to look at it is this... power out = thrust x free_stream_velocity. The power out can never be more than the power in. With a tail-wind, the free_stream_velocity is the vehicle_velocity - the speed of the tail wind. So a given power can give me more thrust in this case.
That is what I was getting at. But I was breaking the first rule of physics: "Never derive in public."
There should be a way to get that result from the thrust equation (which is where I think it ultimately comes from), and it will probably involve breaking each velocity into it's component winds. There will be true wind, and wind induced by the motion of the card. Together these will combine to make a relative wind, but that relative wind will obviously change depending on the tailwind. This is where I predict we will find our answer.
spork":2m1ktvzp said:
The power available however, has nothing to due with true or relative winds. The power available is related to the speed of the cart (which hasn't changed), not the wind speed.
Yes, the power available is equal to the speed of the cart times the retarding force on the wheels from "turning the generator".
If I have increased thrust because of a tailwind, that means I need less input power to the prop to achieve the same thrust as was achieved in the NO TAILWIND case. But I still have the same power available.
Exactly!
This means I have excess power in the TAILWIND case! I will end up accelerating!
Right again.
My top speed will now be limited by my ability to utilize this excess power. The more efficient I make my machine, the higher the possible speed. This has become an engineering problem. Note that this is not perpetual motion.
Right on all counts.
We both agree that this is what the real world case is (or at least I am taking your word on it). I am just trying to figure out why it is the case. I think that was also origin's issue. Without a good "why" it is hard to accept.
spork":2m1ktvzp said:
As we get faster and faster, the tailwind effect will decrease.
I don't think this is true - but perhaps in a relative sense.
At some point the internal resistance of the system will overwhelm it and we will have reached an effective no-tailwind condition. At that point the system will be in equilibrium again.
This is right except for the "effective no-tailwind".
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I stated that poorly I guess. The tailwind is only one component of the relative winds experienced by the system. At some point, even in the theoretical "ideal" model, those other components will eventually overwhelm the effect of the tailwind. It will still be there and won't have decreased. But as you get going faster you will have diminishing returns until it become negligible. This is entirely wild conjecture on my part (just to be clear).
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