Mathematical Analysis of an Ideal ZED

[mondrasek]

No.  You are mixing circumstances of compressible and non-compressible substances.  Work is always the integral of F*ds.  If we take a capsule of fluid and subject it to 1psi or a million psi we have not done any work on that fluid.  If we apply pressure against a cross section of fluid through a distance, then we do work.  When we lift columns of fluid we can obtain the work performed and stored by solving the F*ds integral which will work out for a single column to:  E=0.5*total_weight*height = pave*volume = 0.5*density*volume*height = 0.5*density*area*height². 

The energy is not stored in compression of the fluid for the simple reason that the fluid is incompressible.  The energy is stored in the gravitational potential of the raised mass.  Larry  asserted that raising some cross-section by 1' to end up with the 1+2+4 configuration "cost only 1/3" of some other configuration.  But it doesn't.  The force went from 0 to 3X what it would have raising an isolated column by 1'.  Identically, the amount of work performed was 3X that required to raise an isolated column by 1'.  The force and the energy both scaled by 3X versus the isolated column.  Had we done the exercise totally emptying the middle column, then the force would have gone from zero to 9X over a 3X stroke.  Kf would still be 3*pWater*area, and the integral would be:  0.5*3*pWater*area*(3²-0) = 27*0.5*pWater*area, IE 27X the energy of raising an isolated column by 1' and 3X the energy of raising an isolated column by 3'.

No, MarkE, you are misunderstanding me.  When I said:

The Integral of Pressure * Volume is Energy,

it was in reference to the analysis of the ZED.  The Volume I was referring to is that which moves into and out of the system.  The input Volume is water.  The output Volume is that which is encompassed by the portion of the outer riser that lifts up above the original start condition height.

Sorry I did not make that more clear.  I realize you are looking at several things at once and I did not point out exactly which case I was making reference to. 

[TinselKoala]

Please do not forget that +incompressible fluids+ do not store energy in volume changes due to applied pressure.

In a spring, when you press on it with a certain force (pressure) the spring compresses (changes "volume") and thus the energy integral is valid. If you press with the same force (pressure) on a concrete block... no (or extremely little) energy is stored in compressing the concrete, its volume does not change due to your applied pressure, so the energy integral in the form you have stated it does not apply, just as the various individual ideal gas laws Boyle, Charles, Gay-Lussac and Avogadro, or indeed the full combined ideal gas law pV=nRT  do not apply to incompressible fluids.

[LarryC]

Correction, noticed that the Pod Lift values wasn't squared.

[MarkE]

No, MarkE, you are misunderstanding me.  When I said:

The Integral of Pressure * Volume is Energy,

it was in reference to the analysis of the ZED.  The Volume I was referring to is that which moves into and out of the system.  The input Volume is water.  The output Volume is that which is encompassed by the portion of the outer riser that lifts up above the original start condition height.

Sorry I did not make that more clear.  I realize you are looking at several things at once and I did not point out exactly which case I was making reference to.

If you stick with integral of F*ds there is no room for confusion.  See for example LarryC's example of 0+3+3 columns versus 1+2+4, picking pressures or average pressures and volumes easily leads to non-physical results, whereas integrating F*ds yields the correct result every time.

[MarkE]

Correction, noticed that the Pod Lift values wasn't squared.

Larry, the height used in a calculation is the height of a single column.  In your Stored energy calculations it looks like you took the water column height, subtracted one and then doubled it. 

You need to track the energy in each of the four water volumes as shown in this picture that I posted previously.