Mathematical Analysis of an Ideal ZED

[mondrasek]

MarkE, thank you for that detailed example.  But I'm not sure how it is useful for the ZED analysis I am trying to perform?

Take a volume where we are going to eject water replacing it with an incompressible fluid...

I cannot see where in my analysis that I am ejecting water and replacing it with an incompressible fluid?  To "charge" the ZED additional water is being added to the volume of water that initially existed in the pod chamber.  That initial volume of water did cause an initial pressure (at the bottom of the pod chamber) that would need to be overcome in order for the additional water to be introduced (again, at the bottom of the pod chamber).  As the water is introduced, the total water "head" in the system increases, and so the pressure of the additional water being introduced must rise to overcome it.

I apologize if I am simply missing how your example applies.  Or possibly you are mixing up webby1's attempts from the RAR thread.  I would appreciate if you could clear this up for me.

I agree from your explanation that since the additional water introduced starts and ends at non zero pressures that my assumption that I could use their average as the solution to that integral may be erroneous.  And I definitely need help with defining the proper equations for this specific ZED model to do that integration properly.  Could you take another look at what I am describing in this setup and let me know if your example is still correct and how?  Or assist with one more specific to the ZED model I am trying to analyze?

Thanks,

M.

[MarkE]

Mondrasek the math is generic and applies where ever you displace one fluid for another with a different SG.  That happens in the different chambers of the ZED.  I have not gone through an analysis of your problem so my guidance remains somewhat generic:

1) Calculate energies at each state in a cycle.
2) Make sure that you calculate energy as the integral of F*ds.
3) Make sure that you define F correctly.
4) Tally the four energies for one complete cycle:  Energy at start, energy added, energy removed, energy at end. 
5) Determine net energy gain or loss as:  Net energy expended = Energy at start + energy added - energy removed - energy at end.
Positive values mean the machine is an energy sink.
Negative values mean the machine appears to produce free energy.
6) Perform sanity checks on each of the states evaluated based on understood physics.

[mondrasek]

MarkE, thanks again for explaining further.  And I see where you are coming from.  But a full energy balance of the internal workings of the ZED is both beyond my current abilities and not of so much interest.  For it to be of interest to me I think I would first need to see some simple indication that something unexpected would be found.  And that is why I settled on this current effort.

I am trying to treat the ZED as a "black box" during this part of my analysis.  And then just see if it acts like a simple ideal hydraulic cylinder by conforming to Boyle's law.  So I think I only need to be concerned with the Energy that crosses the boundaries of the "black box" as represented by the integral of PinVin (water pumped into the pod chamber to charge the ZED) and the integral of PoutVout (rise of the ZED due to the buoyant forces caused by the charging).

You have pointed out an apparently valid flaw with the way I was calculating the Pin.  That pressure was not starting or ending at zero.  So my simplification of using the average pressure as the integral value would not be correct.  And that can be resolved by two different methods I believe.  First, the proper integral equation could be used.  Second, the model can be revised to have an initial starting condition where Pin is zero.  Since the former would require much assistance and would result in an overall more complex analysis, I think it would be best if I tried the latter.

Thanks again!

M.

[MarkE]

Mondrasek, I think it is fine to try and take a greatly simplified view of the box.  If you get an answer that makes sense then there is a decent chance that it is reasonable.  If you get an answer that doesn't make sense such as seems to show over unity, then it is time to look more closely.

[mondrasek]

All,  I need someone to double check my work.  I have redrawn the model so that the start condition has no water in the pod chamber.  AFAIK this model is expected to be bound by Boyle's law (P1V1 = P2V2) under the ideal conditions being analyzed.  But it does not calculate to do so.  I am again finding PinVin > PoutVout, but by a much smaller margin.  I've triple checked all my calcs, so unless I am missing something simple, either a) the analysis process being applied is wrong, or b) the ZED is NOT acting like a simple ideal hydraulic cylinder.

Would anyone care to independently verify my work?  It is fairly simple algebra, no calculus needed.

Thanks,

M.