How Many Pores and What Sizes for 20 ppi Porosity in a Metal Foam?

[blalien]

Homework Statement

This should be a pretty simple question but I can't find a straight answer in the literature. I want to simulate a 3D model of a metal foam by starting with an aluminum block and filling it with randomly placed spherical holes of constant volume. The foam should have a porosity of 20 ppi (pores per inch) and a volume ratio of 25% aluminum. How many pores should there be, and what are their sizes?

Homework Equations

20 pores per inch
25% aluminum, 75% void by volume

The Attempt at a Solution

I'm not clear on precisely what "ppi" means. Does 20 ppi imply that a 1 inch cube of aluminum should contain 20^3 = 8000 pores? And that each pore should have a volume of 0.75/8000 = 0.00009375 in^3? This seems pretty obvious but I want to make sure before I proceed. Thank you for your help!

 

[CWatters]

Assume the pores are spherical. Work out what the radius of each pore would be. Can you put them on a 1/20" pitch/grid without them intersecting each other :-)

I'm afraid I'm not familiar with packing theories.

 

[CWatters]

Google suggests Cubic Close packing can achieve 74% packing density but I'm not quite sure how you calculate the PPI in that configuration..

http://mathworld.wolfram.com/SpherePacking.html

 

[blalien]

The pores in a foam are allowed to overlap with each other. This isn't a sphere packing problem though. This is a question of the expected value of the volume these pores will occupy.

 

[NascentOxygen]

Does 20 ppi imply that a 1 inch cube of aluminum should contain 20^3 = 8000 pores? And that each pore should have a volume of 0.75/8000 = 0.00009375 in^3?

Yes, that is how I see it. As far as averages go, it should make no difference whether the pores are neatly arrayed in equi-spaced rows and columns, or randomly positioned. So, take the easy route, consider them all neatly arrayed in 3D ranks.

Work out the radius each would have if it were spherical, and compare this dimension with their centre-to-centre spacing to see whether the model seems realistic for your application.

 

[CWatters]

If the pores overlap each other won't the volume ratio be incorrect. (eg some of the volume is double counted).

PS I believe they do overlap.

 

[NascentOxygen]

Will the expected overlap be likely to alter the prescribed composition of 25% by more than ±1 sig fig?

 

[CWatters]

If I did the sums right the volume of a single pore after deducting the 6 overlaps is

8.86 * 10^-5 cubic inches compared to
9.38 * 10^-5 calculated by ignoring the overlap

8.86 * 10^-5 * 8000 = 0.71

eg 71% rather than the 75% requested.