Mathematical Analysis of an Ideal ZED

[mondrasek]

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.

Sorry to put you through that explanation.  My question was supposed to be tongue in cheek.  In this case (assumed linear pressure rise starting at zero Pa) that Integral resolves to P_{in average} * V_in.  When solving for the supposed V_out using an Energy Balance the equation is of the form similar to P_inV_in = P_outV_out.  Which is also a representation of Boyle's law.  But just because it has the same form as Boyle's law does not mean it is correct to site that law for this case.  I was incorrect to do so.

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.

When the ZED is released from it's position in the "charged" state, the pod and risers will rise.  When that is happening, the water levels in the various annuli begin to change.  This affects the head level on each of those moving members and their buoyancy changes as they move.  It does not remain constant.  And so the pod and risers will rise until they are in a position where the sum of the buoyancy forces on each individual member resolves to zero.  This is what I am trying to represent when I describe the system rising until it is neutrally buoyant.  Neither the pod or any of the risers actually resolve to an individual neutrally buoyant state in the analysis being performed so far.  In the final state shown in the 2-layer analysis the outer riser is actually negatively buoyant, by a lot!  But the pod and inner riser are still pushing up.  The sum of these forces is a net negative buoyancy and would require that the system sink back down in order to achieve the neutrally buoyant system condition required for no further motion.  And since that appears to require less energy than predicted by the input, I conclude that the energy balance for the 2-layer ZED is showing underunity.

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?

Of course the riser moves a much shorter linear distance than the syringe plunger, assuming a syringe with diameter smaller than the ZED's outer riser.  That is why I proposed to compare volumes.

The work performed by the lift is again the Integral of P_out * V_out.  The Integral of P_out is the average of the starting P due to the buoyant lift force (applied to the outer riser cross sectional area) and the ending P which must be zero.  V_out must equal V_in if the ZED is acting identical to a simple ideal hydraulic cylinder.

[MarkE]

Mondrasek, yes the levels are going to rearrange.  Ignoring the base there are four things that matter:

1) The surrounding atmosphere, and also trapped fluid.  We have defined this to be massless, incompressible material.
2) The riser cylinders.  We have defined these to be massless SG=0 materials.
3) The water.  SG=1.  I don't know what you are using for acceleration due to gravity, or density of water.  I use 9.80665N/kg for G0, and 0.9982g/cc for the density of water at 20C.

The cylinders do not exert any downward force of their own.
The trapped fluid does not exert any downward force due to weight.
Only the water exerts a downward force due to its weight, and a counter upward force on whatever displaces some volume of same.  That is the buoyant force.  Unless you change the problem by adding weights to the cylinders, only the water can be neutrally buoyant.

[mondrasek]

3) The water.  SG=1.  I don't know what you are using for acceleration due to gravity, or density of water.  I use 9.80665N/kg for G0, and 0.9982g/cc for the density of water at 20C.

MarkE, I had used values of 9.81N/kg and 1.0g/cc for simplicity.  I was doing some cross checking with a conversion program that used those values and so also used their roundings for this simple study.  Feel free to use your values or change them as suits your preference.

[mrwayne]

Hello Monderask,

I can remember when you were quite opposed to the ZED system, you were almost hostile - but you helped another engineer "Do the Math" and you asked me very hard questions.

I impressed with your intelligence and character, you did the math.

Our systems do not defy the math - and you are doing a great job presenting that.

Logically, that is obvious - a person should be able to prove or deny with the "math".

The right questions have to be asked - and the wrong prejudices have to be put on hold.

I hope you are able to teach others - You have certainly earned my respect.

Wayne

[MarkE]

Hello Monderask,

I can remember when you were quite opposed to the ZED system, you were almost hostile - but you helped another engineer "Do the Math" and you asked me very hard questions.

I impressed with your intelligence and character, you did the math.

Our systems do not defy the math - and you are doing a great job presenting that.

Logically, that is obvious - a person should be able to prove or deny with the "math".

The right questions have to be asked - and the wrong prejudices have to be put on hold.

I hope you are able to teach others - You have certainly earned my respect.

Wayne

Isn't it funny then that you have shown neither math that works, nor a unit that works in six years of selling this snake oil?