Mathematical Analysis of an Ideal ZED

[LarryC]

If you stick with integral of F*ds there is no room for confusion.  See for example LarryC's example of 0+3+3 columns versus 1+2+4, picking pressures or average pressures and volumes easily leads to non-physical results, whereas integrating F*ds yields the correct result every time.

MarkE,


Yes, they both end with same stored energy, but you're not considering Power or rate of doing work.


With a pump that is rated for 1 cubic foot per minutes. Then the single column would take 3 minutes and the multiple connected columns would take 1 minute to create the same PSI in each system.

[LarryC]

Larry, the height used in a calculation is the height of a single column.  In your Stored energy calculations it looks like you took the water column height, subtracted one and then doubled it. 

You need to track the energy in each of the four water volumes as shown in this picture that I posted previously.

MarkE,


In your multiple connected column example you used 4-2+1. The Zed is a multiple connected column based system, so why wouldn't it be Riser Head - Riser Gap Head + Pod Head.


Don't understand the 'doubled it' comment. I'm summing the fields and the -value is only used once.

[TinselKoala]

Yes, they both end with same stored energy, but you're not considering Power or rate of doing work.

Power is not energy. Pressure is not energy. Flow rate is not energy.

Where are the sausages (excess energy that can be used outside the system without making the system come to a stop)?

[MarkE]

MarkE,


In your multiple connected column example you used 4-2+1. The Zed is a multiple connected column based system, so why wouldn't it be Riser Head - Riser Gap Head + Pod Head.


Don't understand the 'doubled it' comment. I'm summing the fields and the -value is only used once.

LarryC, energy is the integral of F*ds.  The force that must be exerted to go from the 3+3 state changes from 0 to pWater*area*3ft of total head because the column in the middle counterbalances the column on the right, so it is the difference between those heads, plus the head that we develop in the left hand column that determines the net weight:  IE force that we lift each increment of distance as we pump water into the left hand column.

[MarkE]

MarkE,


Yes, they both end with same stored energy, but you're not considering Power or rate of doing work.


With a pump that is rated for 1 cubic foot per minutes. Then the single column would take 3 minutes and the multiple connected columns would take 1 minute to create the same PSI in each system.

Larry, pressure is no measure of work or power.  I can create lots of pressure instantly without doing any work.  In column K you labeled values as Stored Energy in Ft. Lbs.  In row 8 you have a value of 329.63.  The corresponding head is 35".  The work to fill a column to 35" is:  Integral F*ds = 0.5*62.316lb/cuft*0.27816sqft*(35/12)ft² = 73.69 ft. lbs.  You've calculated a value more than four times that.  You need to calculate energy for all four parts that the drawing identifies: