Mathematical Analysis of an Ideal ZED

[mondrasek]

Of course.

*sigh*  Would you be so kind as to describe the correct method in the form of a mathematical relationship?

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.

Please refer to the start condition and the charged condition shown in the first two diagrams here:

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

A specific volume of water is being added to the pod chamber (AR1).  If the ZED is acting strictly like a simple ideal hydraulic cylinder, then the outer riser should need to rise by an amount that would be equal to having received that same volume of water.  That volume is calculated by measuring how much the outer riser lifts, I think.  It is the same way we correctly measure your U tube diagram:  Draw a box around the ZED.  Then see what enters and exits that box.  Since we are using incompressible fluids the volume of water entering "the box" must be equal to the volume of the outer riser that exits "the box."  Is that correct?

When the charged ZED is released to rise, it will rise due to buoyant force until it is neutrally buoyant.  If the above paragraph is correct, that rise amount should be the measured volume of the outer riser that lifts up (and out of "the box").  So the actual rise * cross sectional area of the outer riser = volume that exited "the box."  That volume must be equal to the volume of water added during the charge.

I propose to draw the ZED as if it has risen to satisfy having stroked by the exact same volume that has been added by the charge.  The position of the fluids will be redistributed correctly due to the constraints of being incompressible.  If the ZED is acting exactly as a simple ideal hydraulic cylinder, then it should come to rest at this condition, and so equalize the buoyant forces caused by the charge and be neutrally buoyant.  If it is found to be NOT neutrally buoyant in this condition, it would have to stroke less or further and definitely not be acting like a simple ideal hydraulic cylinder.

[MarkE]

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

Thanks.  That helps.  37mm will not cause an underflow / overflow problem.

[MarkE]

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?

The work calculation is still performed as the integral of F*ds.  You can normalize by using the area of the feed pipe ID.

[TinselKoala]

*sigh*  Would you be so kind as to describe the correct method in the form of a mathematical relationship?

Say you have a syringe full of water, connected by a tube to the input port of your device. You use a spring scale to press the syringe plunger into the syringe. You monitor the distance the plunger has travelled (s), you record the instantaneous force readings from the scale (F), and you calculate the integral of the spring scale's instantaneous force readings over the distance the plunger has travelled. (Integral from 0 to s of F ds.) This results in an input energy (or work) value. You can press the empty syringe the same distance to get a value for the syringe frictional work that isn't injected, subtract the latter from the former, and you then have the actual work injected into the system.
I thought we had already covered that.

Please refer to the start condition and the charged condition shown in the first two diagrams here:

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

A specific volume of water is being added to the pod chamber (AR1).  If the ZED is acting strictly like a simple ideal hydraulic cylinder, then the outer riser should need to rise by an amount that would be equal to having received that same volume of water.  That volume is calculated by measuring how much the outer riser lifts, I think.  It is the same way we correctly measure your U tube diagram:  Draw a box around the ZED.  Then see what enters and exits that box.  Since we are using incompressible fluids the volume of water entering "the box" must be equal to the volume of the outer riser that exits "the box."  Is that correct?

Maybe. Without knowing exactly what you mean I'm not sure. Recall that the surface area of a cylinder isn't cut in half when you cut the cylinder itself in half across its height, since the ends are still the same area as before... only the "wall" area has been cut in half.

When the charged ZED is released to rise, it will rise due to buoyant force until it is neutrally buoyant.

Nope. The buoyancy of an item does not change as it rises; if it was buoyant at the start, it will rise until the mass of the displaced water is equal to the _entire mass_ of the object.  Neutral buoyancy means that the entire volume of the item displaces the same mass of water as the item itself masses.  If you now only want to count the part of the item that still remains under the water line in your "neutral buoyancy"... I don't think this is legitimate.

If the above paragraph is correct, that rise amount should be the measured volume of the outer riser that lifts up (and out of "the box").  So the actual rise * cross sectional area of the outer riser = volume that exited "the box."  That volume must be equal to the volume of water added during the charge.

Maybe. How _far_ does the riser need to rise in order for that to be true? More, or less, than the distance you depressed the plunger of the syringe? Much less, I'll wager, since the syringe is of smaller cross sectional area. How much work is performed by that lift, though?

I propose to draw the ZED as if it has risen to satisfy having stroked by the exact same volume that has been added by the charge.  The position of the fluids will be redistributed correctly due to the constraints of being incompressible.  If the ZED is acting exactly as a simple ideal hydraulic cylinder, then it should come to rest at this condition, and so equalize the buoyant forces caused by the charge and be neutrally buoyant.  If it is found to be NOT neutrally buoyant in this condition, it would have to stroke less or further and definitely not be acting like a simple ideal hydraulic cylinder.

Automatic bollard. Is the retracted 300 pound bollard nearly "neutrally buoyant" because it only takes a few ounces lift to raise it all the way up? Compressible fluids store energy like springs, in the compression. Incompressible fluids can only store energy by "head height". But when you have raised up the head height of an incompressible fluid you have stored energy there, which can be returned, just as the spring in an automatic bollard helps you lift it easily and lower it gently. Once the bollard is "precharged" by assembling it with its compressed spring, you can lift and lower over and over again as many times as you like, making complete cycles, with only a few ounce-feet of work put in each time. And this is a single-layer system! So you've input a small amount of work (say one pound x lift distance) by reducing the input required to perform the task of lifting the bollard, and you've recovered a huge amount of work (the 300 pounds x lift distance) and so your "net" is just the difference between the two, say 299 pounds x lift distance. Right? So now all you have to do is take some of this "net" and use it to power another automatic bollard. And _THAT_ is where the difficulty lies.

[LarryC]

In the JPG below, I copied some of the important results from the 2 Flow Analysis and did comparisons to highlight some important performance differences between the Zed and Archimedes single and dual versions.


In the original 2 spreadsheets I added a separate column for the important 'PSI differentials' values (peach color), they are key to the efficiency of the Zed. It was previously part of the Input Ft Lbs formula.


Any ideas, questions or suggestions to improve the comprehension of the Zed process with these spreadsheet would be appreciated.


[size=78%]Thanks, Larry [/size]