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### Author Topic: Mathematical Analysis of an Ideal ZED  (Read 761679 times)

#### LarryC

• Hero Member
• Posts: 911
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #150 on: February 27, 2014, 12:30:33 AM »
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

• Hero Member
• Posts: 911
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #151 on: February 27, 2014, 02:10:02 AM »
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

• Hero Member
• Posts: 911
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #152 on: February 27, 2014, 02:34:17 AM »
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

• Hero Member
• Posts: 6830
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #153 on: February 27, 2014, 02:55:22 AM »

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

• Hero Member
• Posts: 911
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #154 on: February 27, 2014, 03:06:02 AM »
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

#### minnie

• Hero Member
• Posts: 1244
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #155 on: February 27, 2014, 08:48:46 AM »

Hi MarkE,
I've been looking at Larry's drawings and don't quite know what "flow assist" means.
I realise you're having to go back over all this stuff and more or less start again and I'm
beginning to sort of understand it.
Thankyou John.

#### MarkE

• Hero Member
• Posts: 6830
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #156 on: February 27, 2014, 08:58:34 AM »

Hi MarkE,
I've been looking at Larry's drawings and don't quite know what "flow assist" means.
I realise you're having to go back over all this stuff and more or less start again and I'm
beginning to sort of understand it.
Thankyou John.
Minnie you will have to ask Larry.  I guess that it means external energy added.  Whenever the machine sloshes water from a high column to a lower column it loses stored energy.  The whole gag has been misdirection by Wayne and company away from calculating energy values correctly over the course of a full cycle.  Please see the next post.

#### MarkE

• Hero Member
• Posts: 6830
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #157 on: February 27, 2014, 09:07:08 AM »
Larry, I've looked at the latest spreadsheet.  There is still a good deal of work to do here.  Please refer to the drawing below:

We need to either insure that the starting and ending energy states are identical, or else account for the stored energy in each state.    No matter what, we do need to calculate the work added and going from each state to the next.  Knowing the stored energy at all states provides a good sanity check.  Please be aware that each time water equalizes from a taller single column to two or more lower columns that we lose stored energy.  The drawing includes formulas for calculating stored energy under the assumptions previously stated:

20C
G0=9.80665m/s/s
zero thickness walls
25" diameter pod
26" diameter pod chamber
51 circular inch riser gap and riser head areas

#### minnie

• Hero Member
• Posts: 1244
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #158 on: February 27, 2014, 01:01:26 PM »

Hi Larry,
could you please explain what is meant by "flow assist".
You show a stop to limit the travel, this would indicate that
there is stored energy to be accounted for.
Thankyou  John.

#### TinselKoala

• Hero Member
• Posts: 13958
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #159 on: February 27, 2014, 01:34:43 PM »
We have Flow Assists on this side of the Red River too.

#### minnie

• Hero Member
• Posts: 1244
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #160 on: February 27, 2014, 03:25:06 PM »

Koala,
I guess Travis has "gone to ground" so to speak.
That is what I imagined flow assist to be!
Love listening to those crickets
John.

#### LarryC

• Hero Member
• Posts: 911
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #161 on: February 27, 2014, 04:03:09 PM »

Hi Larry,
could you please explain what is meant by "flow assist".
You show a stop to limit the travel, this would indicate that
there is stored energy to be accounted for.
Thankyou  John.

Hi John,

Flow assist is the input energy required above that supplied by the differential PSI between the 2 units.

Yes, there is stored energy.

Larry

#### LarryC

• Hero Member
• Posts: 911
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #162 on: February 27, 2014, 05:32:30 PM »
The multiple connected columns require 1/3 the input Ft Lbs as the single column to have the same PSI.

Wouldn't it be great if someone could design a system to utilize and maintain the PSI in the multiple connected columns example.

Oops.

#### MarkE

• Hero Member
• Posts: 6830
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #163 on: February 27, 2014, 07:35:53 PM »
The multiple connected columns require 1/3 the input Ft Lbs as the single column to have the same PSI.

Wouldn't it be great if someone could design a system to utilize and maintain the PSI in the multiple connected columns example.

Oops.
Oops is right Larry.  You miscomputed the input work.  Work is the integral of F*ds.  The initial force adding your 1ft of water is zero.  But the force at the end is:  4+1-2 = 3*0.65psi/ft.  The added work is therefore the integral evaluation from 0 to 1ft of:  0.5*3*0.65psi*area/ft*z2 = 0.5*3*0.65psi*area/ft*1ft2, which happens to be identically the difference between the starting and ending energies of:  EINITIAL = 2*0.5*0.65psi*area*3ft2 = 18*0.5*0.65*area and EFINAL = 0.5*0.65psi*area*(12 + 22 +42) = 0.5*0.65psi*area*21.

#### LarryC

• Hero Member
• Posts: 911
##### Re: Mathematical Analysis of an Ideal ZED
« Reply #164 on: February 27, 2014, 09:07:20 PM »
Oops is right Larry.  You miscomputed the input work.  Work is the integral of F*ds.  The initial force adding your 1ft of water is zero.  But the force at the end is:  4+1-2 = 3*0.65psi/ft.  The added work is therefore the integral evaluation from 0 to 1ft of:  0.5*3*0.65psi*area/ft*z2 = 0.5*3*0.65psi*area/ft*1ft2, which happens to be identically the difference between the starting and ending energies of:  EINITIAL = 2*0.5*0.65psi*area*3ft2 = 18*0.5*0.65*area and EFINAL = 0.5*0.65psi*area*(12 + 22 +42) = 0.5*0.65psi*area*21.

MarkE,

Our Zed spreadsheet output formulas are F*ds as Force * Stroke. The input Ft Lbs formulas use P average * Volume of the fluid moving into the Pod retainer. That also increases the head in the risers.

The pressure rises faster than would account for just the input volume height, due to the riser head change. Are you now saying we have to account for the riser head change? That would be double dipping as its already factored in P average.

Larry