Mathematical Analysis of an Ideal ZED

[LarryC]

Larry, both the ZED and the "Archimedes" scheme rigidly conform to Archimedes' Principle. 

Hi MarkE,


Attached is the 1 Zed to 1 Archimedes spreadsheet. I decided to change the Archimedes water head so that its input Ft Lbs is equal to the Zed input Ft Lbs.


The Zed output Ft Lbs is 33.55% greater than the Archimedes, thus this excess needs to be resolved to rigidly conform.


On the remarks about incompressible air, which causes a 1X head change in the pod area to create a 2X head change in the riser. In the current design the air still compresses but the mechanical changes can produced > 2X. So not an issue.


The Archimedes part (Pod) of the Zed main purpose is to keep the air gap between the Pod and the Riser from increasing. Any additional lift by the Pod is only bonus.


Larry

[LarryC]

Larry, OK I fixed that.  Because the stipulation is equal cross-section area in each column, I changed the formulas to derive the diameters with the areas fixed at 51circular inches, IE 51*pi/4.  Consequently, the formulas are:

Riser area = (26^2 + 51 )*pi/4  = 570.9844647900
Vessel area = (26^2 + 2*51)*pi/4  = 611.0397711233
Riser diameter = (26^2 + 51 )^0.5 = 26.96293752543
Annular clearance = (26.96293752543 - 26)/2 = .4814687627128
Riser diameter = (26^2 + 2*51 )^0.5 = 27.89265136196

The precision of these numbers are not particularly significant once we defined the annular cavities to all have the same 51 circular inch areas and perform our calculations based on that stipulated area rather than calculated area.

MarkE,


Thanks for the calculations. Nice, I understand and learn.


The riser head is still incorrect, but it may be due to my previous explanation. The bottom arrow should be at the same level as the water level height between the riser wall and pod retainer wall.


Larry

[LarryC]

Larry, 28^2/26^2.  Actually, I slipped and the area error is 8%, still that is 40X the 0.2% you think resulted, so that should raise suspicion right there.  The annular ring areas are the differences of squares, so by proportion using your original numbers:  28dia - 27dia = 55cir_area versus 26dia - 25dia = 51cir_area:  55/51 ~8%.

Another point you made, that I said I would answer later. I see how you came up with 8%, but it has little relevance to the efficiency when dealing with pressure differential between 2 Zeds. FYI, I am working a clearer explanation for pressure differential energy calculations.


Larry


 

[MarkE]

Another point you made, that I said I would answer later. I see how you came up with 8%, but it has little relevance to the efficiency when dealing with pressure differential between 2 Zeds. FYI, I am working a clearer explanation for pressure differential energy calculations.


Larry

Larry, if you are about to make significant changes to your spreadsheet, then I will hold-off taking the time to go through the one you just sent. 

[LarryC]

Larry, if you are about to make significant changes to your spreadsheet, then I will hold-off taking the time to go through the one you just sent.

No effect to the one I just sent as it has no differential. It would only effect a new 2 Zed and 2 Archimedes version.


But, no rush.


Thanks, Larry