Mathematical Analysis of an Ideal ZED

[mondrasek]

The Theory Under Test in this Analysis is that the ZED is satisfying an Energy in = Energy out relationship.  Since the Energy in was supplied as a known volume of water injected at a start Pressure of zero, and ending at a higher end Pressure, the Energy in can be calculated as the integral of F*ds, which for this case resolves to simply P_ave*V.

In order to find P_ave it is necessary to find P_end.  P_end is calculated by finding the total water Head at State 2, which creates the Pressure at the bottom of the pod chamber.  The total Head = 175.6364mm of water.  This equates to a Pressure of 1722.9935Pa.

E_in = 0.5 * (P_end) * V = 0.5 * 1722.9935Pa * 2.44102cm³ = 2.10293mJ.

This is also the difference in the Energy in State 2 and Energy in State 1 if you take those numbers from the sum of the Energies due to the water columns in each annulus that are tallied at the bottom of the previously attached spreadsheet.

E_out can be calculated by the same equation as E_in.  But a simpler reduction of the integral of F*ds for this case is F_ave*S where S is the lift distance.

In order to find F_ave it is necessary to find F_start.  F_start is calculated by finding the sum of the buoyant Forces on the pod and risers.  That process was explained several posts ago when on Sunday I began to work on the previous Analysis.  The results of the buoyant Forces for the current SUT at State 2 are:

pod lift             =  11.63548 grams of Force
inner riser lift   =  35.63549 grams of Force
middle riser lift =  45.34175 grams of Force
outer riser lift   =  55.07041 grams of Force
------------------------------------------------
Total Lift          = 147.67154 grams of Force

E_out = E_in = 2.10293mJ = 0.5 * 147.67154g * S.  Solving for S results in a predicted lift of 1.90938mm.  This is the lift shown in State 3.

[mondrasek]

State 3 is drawn with the pod and riser lifted 1.90938mm.  The water in each annulus is distributed properly.  I will not go into all the detail on that, but the necessary values for volume changes within the annuli due to the lift, and the resultant water level changes in each are in the right hand column of values in the spreadsheet.  The results were also double checked by evaluating the volumes of the water and air in each location and comparing to those in State 1 and State 2.  This was done throughout the Analysis at each step since a mistake on any one cascades into the following.  So, while tedious, performing the volumetric double check at each step does save me some rework in the end I believe.

The Hypothesis Under Test is, again, that Energy in = Energy out = 2.10293mJ.  In State 3, the ZED has been allowed to rise so that 2.10293mJ of Energy has been released.  This is Energy that could have performed Work.

If E_in = E_out then the system must be at rest at State 3.  It must have a sum of buoyant Forces on the pod and risers that is zero.  Evaluating the buoyant Forces:

pod lift            =   8.16761 grams of Force
inner riser lift   = 19.74154 grams of Force
middle riser lift = 11.30235 grams of Force
outer riser lift   = -7.38391 grams of Force
------------------------------------------------
Total Lift          = 31.82759 grams of Force

Since the sum of the buoyant Forces at the State 3 predicted by E_in = E_out is not zero the ZED cannot remain in this position.  It must Stroke further.  So additional Energy must exist in the system.

This is how I originally concluded that this system is displaying over unity.

[mondrasek]

The Energy inside the ZED at each step was added to the spreadsheet as an additional triple check.  It also shows that Energy increases when transitioning from State 2 to State 3.

Energy at State 2 = 5.52257mJ
Energy Released (Work) from State 2 to State 3 = 2.10293mJ
Energy at State 3 = 3.74510mJ

The total Energy released and remaining in the system when transitioning from State 2 to State 3 is 2.10293mJ + 3.74510mJ = 5.84803mJ.  This is larger than the amount of Energy in the system at State 2 by .32546mJ.  This also shows the Energy increase from State 2 to State 3.

[LarryC]

Attached is a picture of a spreadsheet and the actual spreadsheet below.


It's shows that doing an Iterative driven Integration, calculating Pressure * Volume for each iteration gives the same results as our use of P average * Volume.


If we were doing an Integration using sensors on a physical system, we couldn't do it that way as each stop and start results in many losses, main ones being momentum and inertial in the Zed. Also, in a physical system, most changes are not linear.


In a small math model like this, it is correct, because it has no losses and the changes are linear. One rule is that you can't use it across multiple SI in one start stop.


I could add the Iterative Integration program to my Analysis of flow spreadsheets, but that would only cause more questions and require more computer time.

[minnie]

  It is quite a problem if the math shows OU. Working from First Principles
  gves us the answer when we start out. Someone said earlier in the
  discussion that "The math doesn't lie".
     I believe the basic principles to be bullet proof, if they weren't you
   couldn't depend on anything.
      To prove some sort of anomaly is going to require an exceptionally
    well done piece of work.
                           John