Regarding A Pneumatic Fraud

[Jonathan]

I've done another analysis, it isn't as encouraging as previous ones though. Also, as I have little experience with thermodynamics, this is more likely than any other to have mistakes.

[Jonathan]

Here is a site about The Newton-Raphson Method. That is pretty much all of it, though they don't mention that for some equations and given starting x's the sequence won't converge.

[Jonathan]

I'm surprised there's not more interest, three of the attachments have been accessed only a handful of times, and the other two are accessed automatically just by viewing this page. I was expecting that someone would've found an error by now, you wouldn't believe the mistakes I found before posting this. I must of made "P(t,dr)&S(t,dr).GIF" five times, because I kept noticing errors in the tank volume equation (of which two turned out to be imaginary upon later inspection).

[Fletcher]

"Gisting" this, you are investigating to your own satisfaction whether JB could have used stored pneumatic pressure to create a fraud.

You conclude that if the technology were available to build suitable pressure storage vessels of sufficient strength & capacity & you were able to fill them to an adequate pre-demonstration pressure using contempory pressure pump systems, then it was conceivable it could have been done but unlikely, mainly because of the complicity required for the rort to succeed & inherent technological problems of coupling & disguise.

-fletcher

[Jonathan]

Yes, but I'm pretty sure the technology to make the tank was available. The primary obstacle to this method of fraud would be concealing it. As I said, no doubt Karl would notice giant pressure tanks, and anyone (including Leibniz who suspected pneumatic fraud) would be able to hear the air that made it turn. (Of course I assume no energy would be lost to sound when calculating how pressurized the tank must be).

[Vic Hays]

Yes, but I'm pretty sure the technology to make the tank was available. The primary obstacle to this method of fraud would be concealing it. As I said, no doubt Karl would notice giant pressure tanks, and anyone (including Leibniz who suspected pneumatic fraud) would be able to hear the air that made it turn. (Of course I assume no energy would be lost to sound when calculating how pressurized the tank must be).

The technology for pressure tanks was available, but not materials for high pressure. Somehow I don't think that a fraud would be pursued so tenaciously.

[trevie]

I've done another analysis, it isn't as encouraging as previous ones though. Also, as I have little experience with thermodynamics, this is more likely than any other to have mistakes.

Jonathan, I have been looking at your tank Volume equation, and I am sure there is an easier way to find the volume of those tubes. without having to be accurate to the ml.

If you can find the area of the circle of one of those tubes and then the length, that will give you the volume. Then work out the same for all the other tubes. To work out the Dome part of each end of the tube, I would Basically work out the volume of the sphere from the radius of the tube.

I have included an example for use in integration. for example to find an area under a curve, very similar to find the volume in a tube that has a curve at either end,

[Jonathan]

The process you describe to find the volume of the tanks is exactly what I did! I didn't use calculus because there are a finite number of tanks. It looks a little more complicated than it is because I don't just specify the radius of the tanks, I specify the diameter and wall thickness, so that the radius is given by
r=(d-2dr)/2
where d is diameter and dr is wall thickness. You'll find this expression in the first equation for V_t given, and if you just replace it with r for brevity, then that equation will be simplified as
V_t=8*(4/3)*pi*(r^3)+2*sum[1,3](pi*(r^2)*L)
The first term is the volume of 8 spheres, and the second term is the volume of 2 sets of 3 different cylinders (remember that L is a function of i, and that the smallest tanks are just spheres).
Also, in your attached paper, is the equation (x^3)+4 or (x^3)+3? The graph has it as 4, but the math has it as 3. For 4, that answer is 28, and 3, it is 26.

[trevie]

The graph should be upto 3 but I drew it to 4. I should of kept it to 3 but it was too late and I thought it wouldnt cause too much confusion.

as for the thinkness of the wall, Is this really irrelevent? being that the tanks are symetrical and you are only trying to find the internal volume and not the volume of space taken for the wheel. unless you really want to know for building reasons.

trevie

[Jonathan]

Yes, it is relevant. Here are the values for tank volume, given the numbers on the original attachment and a wall thickness of 1cm; the first one uses the given equation (V_t1), the second one assumes no tank wall thickness (V_t2), and the last part is the percent error from assuming zero wall thickness:
V_t1=2.8418m^3
V_t2=3.1253m^3
%E=9.98%

[trevie]

I can see where you are coming from now.. Just needed to read through the Formula in the original post again with what you are telling me from your last post..

[Nenamas]

This stuff is so far above my head its not even funny. lol, all these formulas that I no clue what hey are about or mean. But hey I like this kind of stuff and I dont have to know all these formulas to find out how the wheel worked.