Mathematical Analysis of an Ideal ZED

[LarryC]

Hi M.,


Your uncompressible air made it easy to do the flow spreadsheet for 2 zeds with 1 riser. I used the rule that 1X head change in the pod creates 2X head change in the riser to calculate the water heads.


Got an efficiency of ~154% with 2 zed units and ~74% with 1 Zed unit.


It makes it obvious that the average PSI differential between the Left and right zed units during water transfer is key to it's efficiency. I included the Archimedes spreadsheet again, compare the PSI differential from each, the Zed is superior.


I did have weight on the pod as it is critical that a pod sink point is specified for the water head calculations.


I recommend that anyone looking at these, understand the Archimedes calculations first, then the 2 Zed version will be easier.


M., please check for errors, thanks. This will be an interesting simple proof if we get it cleaned up.


Larry[size=78%]   [/size]

[TinselKoala]

No TK, I am just calling you out (regrettably).  Are you, or are you not, using the pseudonym of MarkE ?

Sure, Rosemary, you have finally tracked me down.

It's hilarious, isn't it? People just cannot believe that there could be _two_ different people both of whom want to see real data that support claims, so they must be the same person posting under two different aliases.

Finally!!!!!!!!!!!!!   We get to the Ideal gas law!  Which is PV=nRT.  Which for Isosthermic cases (ie. T₁=T₂) results in  Boyle's law:  PV_input=PV_output.

Boyle, Charles, Gay-Lussac, Avogadro all lumped together.
Check the Wiki article for the derivation.
Unfortunately none of these apply to _incompressible fluids_.

Is the ZED system an "unsealed buoyant object" or not?

M.

Not, as you have stipulated the Zed contains only incompressible fluids. So it is not comparable to the unsealed buoyant object like the Cartesian Diver, whose buoyancy depends upon the Volume of the compressible gas contained within it, which is open to the pressure of the surrounding incompressible fluid. Changing the pressure of the outer incompressible fluid changes the pressure _and hence the volume_ of the compressible gas within it, thus making the Diver rise or sink as its buoyancy is changed by the changes in the pressure of the surrounding incompressible fluid. Your Zeds, ex hypothesi, have only incompressible fluids, so no pressure changes can result in any volume changes anywhere. In an incompressible fluid, volume is conserved. The only way to get more or less volume is physically to add or subtract more of the incompressible fluid.

Gas laws do not apply to incompressible fluids! If a substance does NOT respect the pressure-volume relationship of Boyle's Law... you can't use Boyle's Law to describe or predict its behaviour!! Ditto for Charles's Law, the combined Boyle-Charles Law, or the full "ideal gas law", since in an incompressible fluid the volume does not change with application of external pressure! By Definition!

So you either must drop references to the gas laws in your analysis and workings, or you must use _compressible gas_ instead of one of the incompressible fluids in your model. I'm arguing for the former, as it will simplify your analysis.


And admit it... you are really Webby, aren't you? And you both post as Red Sunset. And you all are on Travis's payroll. Right.

[mondrasek]

I finally got to spending some time with your model today and have a couple of questions.  The first is that when you admit water you want to fill up the pod chamber to 60mm height. 

MarkE, the volume of water admitted is only enough to fill the pod chamber to 37mm height.  That is shown in the second diagram here:

http://www.overunity.com/14299/mathematical-analysis-of-an-ideal-zed/msg387854/#msg387854

[mondrasek]

So you either must drop references to the gas laws in your analysis and workings, or you must use _compressible gas_ instead of one of the incompressible fluids in your model. I'm arguing for the former, as it will simplify your analysis.

Do you (or anyone else) know how to calculate the amount of energy that crosses into a system when a specific volume of water is introduced, starting at a pressure of zero and building linearly to a final pressure of P_in?

Here is what I would propose to try next:  Using the incompressible fluids in the model.  Is it correct to say that the Volume input (volume of water admitted into the bottom of AR1) should be equal to the Volume output as measured by the height change of the outer riser * the cross sectional area of the outer riser?

If so, a ZED that is drawn with that exact amount of rise and with the fluids re-distributed correctly should be neutrally buoyant, right?  If it is acting exactly as a simple ideal hydraulic cylinder?

[TinselKoala]

Do you (or anyone else) know how to calculate the amount of energy that crosses into a system when a specific volume of water is introduced, starting at a pressure of zero and building linearly to a final pressure of P_in?

Of course.

Here is what I would propose to try next:  Using the incompressible fluids in the model.  Is it correct to say that the Volume input (volume of water admitted into the bottom of AR1) should be equal to the Volume output as measured by the height change of the outer riser * the cross sectional area of the outer riser?

So it would seem, if I am understanding you correctly. However, consider the simple lever. If I "admit" a certain weight on the long end and it sinks, should that be equal to the Height Output as measured by the height change of the weight on the short end? I think nested hydraulic cylinders can act like a compound lever, and I think that the multiple layers might distribute an initial "volume input" over several outer risers, so the final "rise" might be small, but with increased force.

If so, a ZED that is drawn with that exact amount of rise and with the fluids re-distributed correctly should be neutrally buoyant, right?  If it is acting exactly as a simple ideal hydraulic cylinder?

Now you have lost me. Neutral buoyancy means that the mass of the displaced water is equal to the mass of the displacing object. Adding additional force pressing down on the "neutrally buoyant" object makes it sink, so is this extra force to be included in the figuring?  Since your risers are "massless" I think you are once again up against a place where your assumptions are non-physical and may be leading you off the correct track.

Try putting some Red Herrings in the water. That has helped Travis and Red Sunset along greatly.