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How to create an algorithms for fitting a network of tubes with a range of diameters into a cube, with maximum volume in the tubes? Assume:

  • 30 cm minimum square side
  • 1 cm minimum pipe diameter
  • 1 mm pipe wall thickness.
  • Cube edge length $L$
  • number of different diameters $n$
  • volume Pipes $= \pi r^2\times\text{length}$.

The tubes form a network in the sense of a continuous network of connected Tubes.

Tricky parts:

  • Finding the ideal number of different radiuses of pipes to use.
  • accounting for curves in the pipes. May be worth assuming straight pipes with U-turns at the ends? Obviously square pipes would be ideal. But we are restricted to circular pipes.

Best to break down the problem into components:

  1. assuming (one) 1 cm diameter pipe, how to fit most volume pipe into 30cm cube.
  2. assuming (multiple) diameter pipes: 1 cm and 5 cm diameter pipes, how to fit most volume pipe into 30 cm cube.
  3. how to determine the optimal number of diameters
    i. given limited number or radii,
    ii. given unlimited number of radii.
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  • $\begingroup$ use the smallest pipes you can, they pack best. You haven't really given any constraints to the length of the pipe, or if there are any turns. en.wikipedia.org/wiki/Circle_packing $\endgroup$ Sep 5 '18 at 19:57
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    $\begingroup$ "they pack best" - not necessarily. You can fit small pipes into the spaces left between bigger pipes. en.wikipedia.org/wiki/Apollonian_gasket $\endgroup$
    – alephzero
    Sep 5 '18 at 20:01
  • $\begingroup$ What's the purpose of this exercise? $\endgroup$
    – mart
    Sep 6 '18 at 8:00
  • $\begingroup$ Ambiguity: Is the minimum diameter constraint internal or external diameter? If it's external then the wall thickness is immaterial. $\endgroup$
    – user6335
    Sep 6 '18 at 9:29
  • $\begingroup$ @Wossname if its external, then the wall thickness still is significant for the calculation of the volume of liquid contained. $\endgroup$
    – Dale
    Sep 7 '18 at 2:13
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If you assume that your pipes are straight, your problem reduces to 2D circle packing as Mohammad suggests. But unlike what he says, using the smallest pipes wont get you the best packing density:

For seven of these radius ratios a compact packing is known that achieves the maximum possible packing fraction (above that of uniformly-sized discs) for mixtures of discs with that radius ratio. The highest packing density is 0.911627478 for a radius ratio of 0.545151042. https://en.wikipedia.org/wiki/Circle_packing

However, in order to write a meaningful algorithm you need to have a minimum diameter constraint, otherwise you will be able to fill the gaps with increasingly smaller tubes.

You may find better responses for this question on math.stackexchange.com

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    $\begingroup$ A critical component that you've omitted here (and is never included in standard circle packing theory) is wall thickness. If you have one choice of diameter, smaller pipes have less wasted space outside the pipes, but a whole lot more wasted space in the metal walls compared to larger pipes. Larger pipes also need thicker walls, but this will be smaller in comparison to the diameter than for small pipes. There is an optimum diameter, but calculating it depends on the material of pipes, pressure they must sustain, heat transfer requirements, etc. $\endgroup$ Sep 5 '18 at 20:50
  • $\begingroup$ True, I assumed that wall thickness is negligible, which may not be the case for this problem. $\endgroup$
    – user190081
    Sep 5 '18 at 20:53

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