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Author Topic: Mathematical Analysis of an Ideal ZED  (Read 646061 times)

mondrasek

• Hero Member
• Posts: 1301
Mathematical Analysis of an Ideal ZED
« on: February 13, 2014, 03:17:30 PM »
All, please check the math.  I would appreciate if you can point out any mistakes in the math, assumptions, logic, and conclusions.  Feel free to send your input by PM if you don’t want to post in the thread.  It would be nice to know if you check the analysis and agree as much as if you find mistakes.

I preformed this analysis using a CAD model, Excel, and an old Casio calculator.  So precision of the values was carried out to as many as 10 significant digits.  I limited the CAD dimensions to only 7 digits after the decimal so errors are introduced but should still give accurate results if rounded to 6 significant digits or there about.

The model is of an ideal 2-layer (2 risers and 1 pod) ZED.  It was constructed using values intended to simplify the analysis, NOT to optimize expected performance or to show OU.

All dimensions in the CAD drawings are in millimeters.

The model was constructed by first starting with a 20 mm diameter x 60 mm tall pod.  Gaps, ring walls, and risers were all constructed by drawing lines that were offset by 1 mm from that pod and subsequent geometry.  Water (white color) was added to heights that approximate a setup condition that was expected to work for the purpose of analysis, but were otherwise randomly selected.

The remaining black areas under and around the risers are filled with air that is ASSUMED to be an incompressible fluid for the purpose of simplifying the analysis.

The risers and pod are first ASSUMED to be weightless.  Due to the position of the water in contact with the inside and outside surfaces of both risers, they both would be neutrally buoyant.  The pod would have an initial buoyant force equal to the weight of the water it is displacing.  This calculates to be ~8.168 grams.  Therefor it is assumed that an equal weight of 8.168 grams is applied to the outer riser in order for this setup to maintain an overall neutral buoyant condition.

Please note that the 1 mm gaps between the upper surfaces of the pod and risers are only present to eliminate any confusion about possible stiction between those surfaces.  In reality there would need to be physical spacers to maintain those gaps.  Those spacers are ASSUMED to be in place and are not shown.

With this construction and set up condition we can calculate the pressure at the bottom of the pod chamber due to the 27 mm of water head resting on it as 264.87 Pa.

More to come.

M.

mondrasek

• Hero Member
• Posts: 1301
Re: Mathematical Analysis of an Ideal ZED
« Reply #1 on: February 13, 2014, 04:40:39 PM »
Now the ZED is “charged” by adding additional water into the bottom of the pod chamber.  This will require Work/Energy that can be calculated by the Integral of the Pressure of that added water multiplied by the Volume of that added water.

The amount of water added was limited to be that which would cause the water level in the pod chamber to riser to 1 mm less than the total height of the pod when the pod and risers are not allowed to rise.  That volume can be calculated to be ~2.1771 cc.  This will be called Vin.

The positions of the air and water around the risers can also be calculated since their volumes must remain constant.  Once those new positions are found the resultant buoyancy force can be calculated.  For the pod the force now will be able to lift ~18.535 grams.  For the inner riser it is ~31.783 grams.  And for the outer riser it is ~40.440 grams.  The total the ZED can now lift is ~90.758 grams.  Since the setup required the outer riser to weigh ~ 8.168 grams, this needs to be subtracted.  The resultant lift capacity is reduced then to ~82.590 grams.

The Pressure of the water added during this charging cycle must rise from the initial pressure of 264.87 Pa (calculated in first post) due to the increase in the water head in the system.  The total water head in the final charged state is ~151.3463 mm and so the final pressure is ~1484.71 Pa.

If we ASSUME that the Pressure rise of the added water was linear, then the Integral of the Pressure for the added water become the average of the initial pressure before charging and the final pressure after charging.  The average of those two values is ~872.2567 Pa.  This will be called Pin.

The total Work/Energy needed to introduce the additional water to charge the ZED is now calculated as Pin x Vin which works out to ~1.8990 x 10^-3 N.m.

M.

MarkE

• Hero Member
• Posts: 6830
Re: Mathematical Analysis of an Ideal ZED
« Reply #2 on: February 13, 2014, 05:35:00 PM »
The first thing that you will want to be very careful with is the assumptions that you make with respect to the amount of work it takes to displace "air" or water in any of the chambers.  A mistake can easily throw energy off by the ratio of area of the cylinder area to the annular ring area of the chamber.  It is tempting to calculate force based on the smaller area in cases where it is actually a function of the much larger area.

TinselKoala

• Hero Member
• Posts: 13968
Re: Mathematical Analysis of an Ideal ZED
« Reply #3 on: February 13, 2014, 05:36:06 PM »
Quote
And yes - TK a single ZED can be OU - if you simple store the recycled energy and return it on the next stroke, as I said two yeas ago - which you omitted every time you miss applied the context. As I said before - why add the extra effort and time - simply transfer between systems. Get over it - you missed it.

Well, that's a relief. You aren't going to need to complicate things by making dual ZEDs, because we here ONCE AGAIN have Wayne Travis claiming that a single ZED can be OU.... "IF".  Maybe he'll deign to give YOU , mondrasek, the design for the "simple, three layer system that is clearly OU by itself" (his own exact words). Don't hold your breath though.

Quote
On another note - my contact information has never changed - to  those that have tried to discredit me and our company all this time - those people never made one call or asked me one single question about their claims against us. Shame on all of you who slander by assumption.

Are you quite sure of that, Travis? How exactly do you KNOW that some of your detractors and scientific critics haven't been there, haven't talked to you and your "engineers" and your lawyers? You shouldn't be so confident, because you DON'T KNOW EVERYTHING that is transpiring around you. Your "contact information" hasn't changed? But your websites certainly have. Name changes, removal of many of the outrageous claims, the Zed animation is gone..... it has been clear for some time that you are scrambling about, reorganizing, shuffling that pea around, flipping the cards in your three-card Monte game. By the way, there is a difference between slander and libel, you should learn what it is.

MarkE

• Hero Member
• Posts: 6830
Re: Mathematical Analysis of an Ideal ZED
« Reply #4 on: February 13, 2014, 05:49:47 PM »
Beware of nondescript old pickup trucks.

mondrasek

• Hero Member
• Posts: 1301
Re: Mathematical Analysis of an Ideal ZED
« Reply #5 on: February 13, 2014, 06:44:09 PM »
The first thing that you will want to be very careful with is the assumptions that you make with respect to the amount of work it takes to displace "air" or water in any of the chambers.  A mistake can easily throw energy off by the ratio of area of the cylinder area to the annular ring area of the chamber.  It is tempting to calculate force based on the smaller area in cases where it is actually a function of the much larger area.

Thanks for the input, MarkE.  Please let me know if you see any of those type of mistakes so far.

mondrasek

• Hero Member
• Posts: 1301
Re: Mathematical Analysis of an Ideal ZED
« Reply #6 on: February 13, 2014, 08:09:18 PM »
If we ASSUME that the ZED is acting exactly as a hydraulic cylinder, then it would have to follow Boyle’s Law.  The Integral of PinVin must equal the Integral of PoutVout.  To find the lift that would result in this case requires that we find the Vout.  Again I must also ASSUME that the output Pressure that the outer riser can provide is a linear function, starting at an initial pressure found from the buoyant lift force applied to the cross sectional area of the outer riser, and ending at a pressure of zero.  The hydraulic lift force of ~82.590 grams results in an initial pressure value of ~795.980 Pa.  The average Pout becomes half of that, or ~397.990 Pa.  With Pin of ~872.257 Pa and Vin of ~2.1771 cc (from a previous post), we solve PinVin = PoutVout for Vout = ~4.771 cc.  The lift of the outer riser is then calculated to be ~4.688 mm.  This is drawn and analyzed (after redistributing the fluids properly) to see if it is neutrally buoyant or not.

The evaluation of this ZED shows it is definitely NOT neutrally buoyant.  In fact, it sucks, literally.  It is displaying a positive lift force from the pod of ~10.050 grams.  But the risers are both negatively buoyant.  The inner riser has a lift force of ~-7.238 grams and the outer riser has a lift force of ~-43.130 grams.  The total is ~-40.318 grams.  When we add the additional downward force of the ~8.168 grams the outer riser needed to weigh for the system to be neutrally buoyant in the setup position, the total lift in this analysis is now ~-48.486 grams, far below a neutral buoyant condition.

So this test failed.  The ZED could never rise to the height calculated by Boyle’s Law.  If the ASSUMPTION of linear pressure transfers are correct (or close) the ZED could only rise a bit less than 2/3 of the required value necessary to satisfy Boyle’s Law.  Therefor we are left with the possibilities that a) there is a mistake in the math, b) the ASSUMPTIONS are greatly skewing us away from expected results, or c) an Ideal ZED is NOT analogous to an Ideal Hydraulic Cylinder.

M.

MarkE

• Hero Member
• Posts: 6830
Re: Mathematical Analysis of an Ideal ZED
« Reply #7 on: February 13, 2014, 08:16:27 PM »
If we ASSUME that the ZED is acting exactly as a hydraulic cylinder, then it would have to follow Boyle’s Law.  The Integral of PinVin must equal the Integral of PoutVout.  To find the lift that would result in this case requires that we find the Vout.  Again I must also ASSUME that the output Pressure that the outer riser can provide is a linear function, starting at an initial pressure found from the buoyant lift force applied to the cross sectional area of the outer riser, and ending at a pressure of zero.  The hydraulic lift force of ~82.590 grams results in an initial pressure value of ~795.980 Pa.  The average Pout becomes half of that, or ~397.990 Pa.  With Pin of ~872.257 Pa and Vin of ~2.1771 cc (from a previous post), we solve PinVin = PoutVout for Vout = ~4.771 cc.  The lift of the outer riser is then calculated to be ~4.688 mm.  This is drawn and analyzed (after redistributing the fluids properly) to see if it is neutrally buoyant or not.

The evaluation of this ZED shows it is definitely NOT neutrally buoyant.  In fact, it sucks, literally.  It is displaying a positive lift force from the pod of ~10.050 grams.  But the risers are both negatively buoyant.  The inner riser has a lift force of ~-7.238 grams and the outer riser has a lift force of ~-43.130 grams.  The total is ~-40.31771688 grams.  When we add the additional downward force of the ~8.168 grams the outer riser needed to weigh for the system to be neutrally buoyant in the setup position, the total lift in this analysis is now ~-48.486 grams, far below a neutral buoyant condition.

So this test failed.  The ZED could never rise to the height calculated by Boyle’s Law.  If the ASSUMPTION of linear pressure transfers are correct (or close) the ZED could only rise a bit less than 2/3 of the required value necessary to satisfy Boyle’s Law.  Therefor we are left with the possibilities that a) there is a mistake in the math, b) the ASSUMPTIONS are greatly skewing us away from expected results, or c) an Ideal ZED is NOT analogous to an Ideal Hydraulic Cylinder.

M.
Something else that one needs to be careful about is evaluating integrals.  Where the pressure which translates to force changes, we need to solve the integral.  If the force or pressure starts at zero and goes to some other value then the integral is trivial.  If the pressure / force starts and ends at non-zero values then we get both linear and quadratic terms.

mondrasek

• Hero Member
• Posts: 1301
Re: Mathematical Analysis of an Ideal ZED
« Reply #8 on: February 13, 2014, 09:36:34 PM »
Something else that one needs to be careful about is evaluating integrals.  Where the pressure which translates to force changes, we need to solve the integral.  If the force or pressure starts at zero and goes to some other value then the integral is trivial.  If the pressure / force starts and ends at non-zero values then we get both linear and quadratic terms.

Can you lend assistance with how to evaluate the PinVin integral properly since it starts from a non-zero condition in this setup?  Or would it be necessary for me to start over with a setup that initially has no water in the pod chamber?

Also, to be clear, is there any issue with evaluating the PoutVout where Pout starts at non-zero but should end at zero?

M.
« Last Edit: February 14, 2014, 12:34:59 AM by mondrasek »

MarkE

• Hero Member
• Posts: 6830
Re: Mathematical Analysis of an Ideal ZED
« Reply #9 on: February 14, 2014, 05:13:15 AM »
Can you lend assistance with how to evaluate the PinVin integral properly since it starts from a non-zero condition in this setup?  Or would it be necessary for me to start over with a setup that initially has no water in the pod chamber?

Also, to be clear, is there any issue with evaluating the PoutVout where Pout starts at non-zero but should end at zero?

M.
Sure:

Take a volume where we are going to eject water replacing it with an incompressible fluid, where:

H is the height of the volume.
A is the cross-sectional area of the volume.
Ge is the acceleration due to gravity on earth.
pW is the density of water.
pX is the density of the incompressible fluid.

The pressure difference from bottom to top of the volume varies from 0 to H*Ge*(pW-pX).
The force required varies from 0 to H is A*Ge*(pW-Px)*H.
The work done is the integral of F*ds: = Integral( A*Ge*(pW-Px)*H dh)
The solution of the integral from 0 to H is of the form:  Kh*(H2^2 - H1^2) + F0*( 0.5*A*Ge*(pW-pX)*H^2

For pX = 0:  = 0.5*A*Ge*pW*H^2
And since the weight of water in the volume would be: Wdisplaced = A*Ge*pW*H, we get:  E = H/2*Wdisplaced.

That should be intuitively satisfying because we "lift" only an infinitesimal amount of water by 0 height at the start, and another infinitesimal amount of water by all of the height at the end, so the average amount by which we "lift" all of the water that we displace is 0.5*H.  So far, so good.

But what happens when we already have displaced some water?  The math is still the same, just some terms are no longer zero.  Force needs to be defined as a function of what is being changed, displaced height for our problems, and then integrated.  The net work done is the difference between the integral at the start and end points.  Typically, this reduces to:

F = F0 + Kf*H  (Kf may be positive or negative depending on the circumstances)
Integral from H1 to H2 is:  F0*H2 + 0.5*Kf*H2^2 - F0*H1 - 0.5*Kf*H1^2.

mondrasek

• Hero Member
• Posts: 1301
Re: Mathematical Analysis of an Ideal ZED
« Reply #10 on: February 14, 2014, 03:43:08 PM »
MarkE, thank you for that detailed example.  But I'm not sure how it is useful for the ZED analysis I am trying to perform?

Take a volume where we are going to eject water replacing it with an incompressible fluid...

I cannot see where in my analysis that I am ejecting water and replacing it with an incompressible fluid?  To "charge" the ZED additional water is being added to the volume of water that initially existed in the pod chamber.  That initial volume of water did cause an initial pressure (at the bottom of the pod chamber) that would need to be overcome in order for the additional water to be introduced (again, at the bottom of the pod chamber).  As the water is introduced, the total water "head" in the system increases, and so the pressure of the additional water being introduced must rise to overcome it.

I apologize if I am simply missing how your example applies.  Or possibly you are mixing up webby1's attempts from the RAR thread.  I would appreciate if you could clear this up for me.

I agree from your explanation that since the additional water introduced starts and ends at non zero pressures that my assumption that I could use their average as the solution to that integral may be erroneous.  And I definitely need help with defining the proper equations for this specific ZED model to do that integration properly.  Could you take another look at what I am describing in this setup and let me know if your example is still correct and how?  Or assist with one more specific to the ZED model I am trying to analyze?

Thanks,

M.

MarkE

• Hero Member
• Posts: 6830
Re: Mathematical Analysis of an Ideal ZED
« Reply #11 on: February 14, 2014, 04:08:32 PM »
Mondrasek the math is generic and applies where ever you displace one fluid for another with a different SG.  That happens in the different chambers of the ZED.  I have not gone through an analysis of your problem so my guidance remains somewhat generic:

1) Calculate energies at each state in a cycle.
2) Make sure that you calculate energy as the integral of F*ds.
3) Make sure that you define F correctly.
4) Tally the four energies for one complete cycle:  Energy at start, energy added, energy removed, energy at end.
5) Determine net energy gain or loss as:  Net energy expended = Energy at start + energy added - energy removed - energy at end.
Positive values mean the machine is an energy sink.
Negative values mean the machine appears to produce free energy.
6) Perform sanity checks on each of the states evaluated based on understood physics.

mondrasek

• Hero Member
• Posts: 1301
Re: Mathematical Analysis of an Ideal ZED
« Reply #12 on: February 14, 2014, 06:47:49 PM »
MarkE, thanks again for explaining further.  And I see where you are coming from.  But a full energy balance of the internal workings of the ZED is both beyond my current abilities and not of so much interest.  For it to be of interest to me I think I would first need to see some simple indication that something unexpected would be found.  And that is why I settled on this current effort.

I am trying to treat the ZED as a "black box" during this part of my analysis.  And then just see if it acts like a simple ideal hydraulic cylinder by conforming to Boyle's law.  So I think I only need to be concerned with the Energy that crosses the boundaries of the "black box" as represented by the integral of PinVin (water pumped into the pod chamber to charge the ZED) and the integral of PoutVout (rise of the ZED due to the buoyant forces caused by the charging).

You have pointed out an apparently valid flaw with the way I was calculating the Pin.  That pressure was not starting or ending at zero.  So my simplification of using the average pressure as the integral value would not be correct.  And that can be resolved by two different methods I believe.  First, the proper integral equation could be used.  Second, the model can be revised to have an initial starting condition where Pin is zero.  Since the former would require much assistance and would result in an overall more complex analysis, I think it would be best if I tried the latter.

Thanks again!

M.

MarkE

• Hero Member
• Posts: 6830
Re: Mathematical Analysis of an Ideal ZED
« Reply #13 on: February 14, 2014, 06:53:33 PM »
Mondrasek, I think it is fine to try and take a greatly simplified view of the box.  If you get an answer that makes sense then there is a decent chance that it is reasonable.  If you get an answer that doesn't make sense such as seems to show over unity, then it is time to look more closely.

mondrasek

• Hero Member
• Posts: 1301
Re: Mathematical Analysis of an Ideal ZED
« Reply #14 on: February 14, 2014, 09:59:30 PM »
All,  I need someone to double check my work.  I have redrawn the model so that the start condition has no water in the pod chamber.  AFAIK this model is expected to be bound by Boyle's law (P1V1 = P2V2) under the ideal conditions being analyzed.  But it does not calculate to do so.  I am again finding PinVin > PoutVout, but by a much smaller margin.  I've triple checked all my calcs, so unless I am missing something simple, either a) the analysis process being applied is wrong, or b) the ZED is NOT acting like a simple ideal hydraulic cylinder.

Would anyone care to independently verify my work?  It is fairly simple algebra, no calculus needed.

Thanks,

M.