Mathematical Analysis of an Ideal ZED

[MarkE]

Went ahead and added a riser gap that would give it the same SI as the pod gap. Wanted to see how big a difference it would make in the efficiency. Darn, it went from 153.94% to 153.56%.
Larry

Larry, that helps.  I will continue to go through the spreadsheet.  Please confirm that the drawing below is correct:

[TinselKoala]

That would suck on so many levels.  But I think we could then resort to labels.  But it would be so much harder to follow.

My apologies to any colorblind individuals trying to follow along.

TK, I never said that the atmospheric air pressure makes any difference to any pressures inside the ZED.  Thankfully, it doesn't!  What having the outer annulus open to the atmosphere does do is allow for the V_in to NOT equal the V_out.  Because air also moves freely into and out of the system from the atmosphere and adds ANOTHER V (that is not costing us anything) that must be accounted for in an Energy Balance.

The air that freely crosses into and out of the system allows the water levels to redistribute to satisfy simple volumetric constraints.  And when allowed to do so it results in a lift force that is due to BUOYANCY, not the usual pressure * volume relationship found in a simple hydraulic cylinder.

Air freely crosses the boundary of your incorrectly drawn Red Box. The air does not enter the Zed system at all. The outer ringwall's only purpose is to keep the two outer incompressible fluid columns from running off down the drain. The level of the outermost liquid layer is partly determined by the pressure--- the _constant pressure_ -- exerted by the air, and this does not change, it does not move in and out of the system. The only thing the air pressure does is to press down on the outer fluid columns, and it presses down with the same pressure, a constant force, no matter the stage of the system. It does not add another "V" .... the volume of the outer air is essentially infinite, anyway, and so it does not change. Now, if you want to build a zed that is a hundred thousand feet tall, things will be (slightly) different wrt the outside air, because then the pressure it exerts on the top of the outer fluid layer will change, from zero altitude to the highest altitude of the water level. But for realistically sized Zeds, the volume and pressure of the outer air is constant, volume infinite and unchanging, and at the local atmospheric pressure. And don't forget, please, that the outer air pressure is also pressing down on the entire top surface of your apparatus, with that same pressure, but over a much larger area than the area of the outer fluid column.

The proper drawing of your red box would follow the outline of the surface of the outer liquid layer, I think, because that is the sealed surface. It rises and sinks, of course, but that's no problem as far as the boundary condition goes.

[TinselKoala]

The interview went well.

Now that I am sure you know the concept of making ignorant claims against other people is wrong - I will apologize to you when you do.

Wayne

Save your apologies for the judge. I understand they take remorse into account when sentencing.


Now.... just what part of "Show me the sausages" don't you understand?

Show me the Sausages!    A philosopher designs a marvellous sausage machine. A scientist comes to marvel at this wonderful creation, and raises an eyebrow.
The philosopher says, "Ah, behold the wonderful cogs and sprockets and temperature-controlled mixing chambers in my wonderful machine - surely you can see how it must produce the most fantastic sausages!"
The scientist says "Yes, that is all very interesting. Show me the sausages."
The philosopher says "How dare you, a mere scientist, question my wonderful philosophical reasoning?"
Scientist: "I'm not questioning your reasoning - I want to know if your machine really produces sausages."
Philosopher: "Can you point to any flaw in my argument that it produces sausages?"
Sci: "I don't know - I just want to know if it produces sausages. Here is some meat. Why don't you feed it through and see if you get any sausages?"
Phil: "And sully my wonderful machine with mere offal?"
Sci: "You said it was a sausage machine. I want to see the sausages."
Phil: "Are you questioning my ingredients?"
Sci: "I'm just questioning whether it produces sausages or not. Show me the sausages."
Phil: "Ah, so you cannot attack my premises and you cannot attack my argument. Therefore I'm right and you lose."
Sci: "Don't be such a melodramatic prancing arse. Show me the sausages."
Phil: "The sausages inevitably flow from the argument. You see my fine machine.  You can even inspect the meat & onions. The sausages necessarily flow."
Sci: "Show me the sausages or I'm off to Tesco."
Phil: "You are a mere scientist with no understanding of philosophical matters."
Sci: "Bye."

http://answersingenes.blogspot.com/2011/06/show-me-sausages.html

You see, Travis? You aren't the only person with no sausages.
Don't be such a melodramatic prancing arse. Show me the sausages.

[mondrasek]

The proper drawing of your red box would follow the outline of the surface of the outer liquid layer, I think, because that is the sealed surface. It rises and sinks, of course, but that's no problem as far as the boundary condition goes.

But that would require that the red box changes shape (edit: but NOT change by an unpredicted volume) as the Energy Balance is performed.

Maybe so.  Could you explain further?

[LarryC]

Larry, it is not my hang up.  It is that the math does not represent the model represented.  It's like estimating pi as 3.  Whether or not that is close enough depends on the circumstances.  In order to determine the magnitude of the error, one has to reverse engineer your spreadsheet, guess your intent, then substitute the correct relationships and evaluate the differences.  That is a big PITA and rather unreasonable.  It would be very helpful for you to state your assumptions, and for you to perform sanity tests on your own as to the validity of those assumptions.  Introducing ~16% error terms is a recipe for trouble.  I don't care if you fix the ring dimensions for constant area or keep them on a 0.5" grid as long as your calculations represent the model faithfully.

Let me make a suggestion that will make it easier to keep simple numbers on the spreadsheet:  Assign a constant to pi/4.  Then you can represent all your circular areas in integer units multiplied by the constant.  This should make it easier for you to audit your calculations.  The other thing that can be an immense help is to use named fields.  That saves a lot of chasing around.  Instead of a formula looking like: = $H$2*F19*E12 it would look like:  = riser_diameter*riser_length.  My last suggestion you may or may not like:  Using MKS units generally makes it easier to avoid mistakes between mass and force.  I can work in whatever units you are comfortable using.

MarkE,


Thanks for the suggestion, some will help.


But, don't understand your ~16% error, the .38% drop in efficiency was a ~.2% error.


PI/4 constant would help.


On the named fields, I do use them all the time for VBA modules. But, It is a good suggestion to use names on the constant parameter fields at the top for this example. I do like to use them at the multiple line level, in this case the 'Cycles', because when you copy one line to the next, the named fields do not increment its position. I use the 'Trace Precedent' to check formulas, it points to all the fields in the formula.


I agree with your points about MKS units. I don't use it because I've worked with many field engineers that use Imperial because the field workers that apply the specifications wouldn't understand and most times upper management wouldn't either. And I am more comfortable using Imperial, so thanks for working with my unit choice. 


Larry