Mathematical Analysis of an Ideal ZED

[mondrasek]

You have seen my older spreadsheet with recursive iterations performing millions of calculations to get the water levels in the risers correct during the rise, due to interconnection of air compression/decompression. I don't know how to do this it any other way, so I can not help with your calculations.

Hi LarryC!

Oh yeah, I remember the spreadsheets and the iterative nature of those calcs!  I even did some by hand a few times but found that too frustratingly tedious.  I would have had to ask someone to write a program just like you did in your spreadsheet if I was to continue that way!  Hopefully MarkE can show us the proper way to simply set up and solve the Integrals instead.

In this "ideal" analysis the air is assumed to be incompressible.  So no iterations needed.  I just solve for all the needed values on a calculator and keep track of them in a spreadsheet and by drawing the results and measuring new results and relationships in CAD.  CAD is not even necessary but saves me some time.  It is all very straight forward math now.  I also use the spreadsheet to do some quick volume checks at each step to double check my work.  If you don't catch a mistake in one fluid interface adjustment it throws off all the subsequent ones!  A complete analysis of the model presented in this thread only takes me about two hours now, including those double checks.

M.

[mrwayne]

Great collaboration! I am logging out till needed.

Looks Like you Men have a great handle on the ZED system.

Wayne

[mondrasek]

I was hoping to have someone check my math and process of analyzing the 2-layer system before ever posting the 3-layer.  That has not happened and so here is the next model if anyone is interested.  It utilizes the exact same 2-layer model and adds an additional third riser on the outside.  That way the same calculations for the 2-layer portion to find the water levels after introducing the Vin volume "charge" could be re-used.

The PinVin I calculate now rises to ~2.103 mJ.  If PoutVout is to be equal to that per Boyle's law, then the system should stroke ~1.9094 mm.  That is drawn on the right hand side and analyzed to see if it is neutrally buoyant.  It is not, and actually is still pushing upward with ~31.8276 grams of force.  So the ZED would stroke further than shown until it could come to rest again with neutral buoyancy.  And that would require that PouVout would be greater than the PinVin of ~2.103 mJ.

M.

[mondrasek]

All joking aside, the system to be analyzed is surrounded by the red box.  Everything that occurred previous to these steps are "manufacturing" or "setup" costs/conditions for the process to be analyzed and do not need to be considered.

There are exactly two "units" (blue box) crossing into the system on the right.  That is equal to the two "units" that are exiting the system on the left.  Both are crossing into or out of the system at the same height and both are moving the same vertical distance.

[TinselKoala]

And your nested system is significantly different.... how?

1. You are confusing yourself with "Boyle's Law" pressure/volume calculations, because Travis and Red Sunset and Webby have all said that the air can be replaced with incompressible fluid. All chambers in your system should be filled with fluid that doesn't change in volume when it's under pressure.

1b. Air -- wet air -- is not an ideal gas and you will find that it deviates from strict Boyle-Charles law behaviour. How much? Enough to account for your numerical result? I don't know. Do you?

2. You should be able to demonstrate some actual gain in something, somewhere, using just three layers. Travis has told us so!

3. What, exactly, is the real "output" of your system? Is it a lifted weight? If you are counting pressing against a stop as "output work", that's not kosher!