Packing ratios within a rectangular box

[swbluto]

How should a person pack cylindrical cells within a rectangular prism(i.e., box) whose width and height perfectly meet with whatever arrangment the cells take(the box can be customized to whatever size needed)? So far, it seems two of the most popular methods include "1 cell directly ontop of the cell below it" and the "pyramidal arrangement", like so:

(one cell directly atop the one below)
O.O.O
O.O.O
O.O.O

(pyramidal type of arrangement)
O.O.O
.O.O.O
O.O.O

(The O's are the cylindrical cells and the periods are just "white space" so I can make the proper shape. :wink: )

Assuming N columns of cells and M rows of cells, the packing ratio for the top pattern is pi/4 and the "pyramidal arrangment pattern" is, for odd numbers of M,..

pi*N*M
-------------------------------------------(dividing line)
[2N+1]*[2M+1+(1/2)*3^(1/2)]

Ideally, for a given N and M, the denomiator should be as small as possible and as N and M go large(and approach infinity), the packing ratio approaches pi/4. For finite N and M, which is where we work at :lol: , the packing ratio is less than pi/4 and decreasingly so for smaller values of N and M.

So in effect, the regular stacking pattern(top one) is the most efficient for a rectangular box. Now, if you can start fitting in parallelogram boxes into wherever(like the diamond of a bike-frame), the pyramidal arrangment would become competitive as the packing ratio could exceed pi/4. Now, who's upto analyzing the ideal packing ratio for the ideal parallelgram case?

 

[Zoot Katz]

I'm not up for the math.

My solution* was found by making a mock up or the proposed box to fit my particular frame and then cutting mock up batteries from the same diameter tubing. (golf club shaft proctectors are within a mm of D, DD and F cells) The cells' lengths are different.

Don't forget the possiblity of stick formations too. Dare I mentions solderless?

*My dummy cells were cut the length of DD cells and then I bought F size cells. It's back to the drawing board, ehh, chop saw for me.
A CAD wiz has more options.

 

[PaulM]

I think I disagree that the rectangular arrangement is more efficient. I didn't do the math, just a quick drawing in AutoCAD. The area of the box for the rectangular is 96, for the parallelogram it is 93.6. The longer and flatter the pack, the more efficient the parallelogram becomes as the number of "dead spaces" per given size is reduced.

 

[swbluto]

I think I disagree that the rectangular arrangement is more efficient. I didn't do the math, just a quick drawing in AutoCAD. The area of the box for the rectangular is 96, for the parallelogram it is 93.6. The longer and flatter the pack, the more efficient the parallelogram becomes as the number of "dead spaces" per given size is reduced.

Then there's probably something wrong with my math then. Thanks for the in-reality double-checking: it seems verification is needed whenever crossing over the mathematical-to-physical bridge.

I wonder, though, if that depends on the even-ness of M. My analysis was carried out with odd numbers of M(the number of rows), but your drawing is an even number of M(4). I wouldn't think the end-results would be that different, though.

 

[paull]

The math got goofed somewhere. This is the formula I get and it should work for any N and M > 1:

M*N*pi
----------------
(2*N + 1)*(2+M*sqrt(3)-sqrt(3))

For the 6x4 that PaulM drew this works out to about 81% packing efficiency vs the standard pi/4=78% efficiency.

 

[swbluto]

The math got goofed somewhere. This is the formula I get and it should work for any N and M > 1:

M*N*pi
----------------
(2*N + 1)*(2+M*sqrt(3)-sqrt(3))

For the 6x4 that PaulM drew this works out to about 81% packing efficiency vs the standard pi/4=78% efficiency.

Okay, so the asymptotic packing efficiency should be pi/(2*sqrt(3)) ~= 90.7% as opposed to the ordinary method's ~=78.5%, with an efficiency greater than or equal to 78.5 for, uhhh, almost all "large packs". Seems like smaller packs should just be directly calculated.

I see that our answers are very similar, except for the right side of the denominator that contains the M term. Hmmmm.... It seems like it's probably my failed geometric reasoning given your M's coefficient is sqrt(3) instead of my 2.

 

[methods]

I have over analyzed a lot of stuff but you are the master =)

Do as Zoot Katz suggests and mock it up. No matter how much math you do in the end real life details will spoil the best laid plans. Things like the thickness of the glue, tape wrapping, wire routing, cooling channels, manufacturing tolerances on both the batteries and the box, changes of plans, etc. always burn me when I try to get too exact.

My best inventions and most satisfying projects have been stumbled upon with a hot glue gun in my left hand and a exacto in my right.

-methods

 

[Ypedal]

agreed ! :wink:

Like trying to stuff 10 lbs of crap in a 5 lb bag !!

No matter how much calculations I take the time to do, i always end up just getting it done hands on style..

 

[docnjoj]

My best inventions and most satisfying projects have been stumbled upon with a hot glue gun in my left hand and a exacto in my right.

-methods
Sorry, but I couldn't resist! That sounds like a recipe for stabbing yourself in the leg with one hand and then gluing it to the other arm as you fall down the stairs
otherDoc