# Is there a relation between porosity and PPI density in porous media?

I've seen two methods to describe porosity of porous media (specifically metal foams):

1. Porosity $\epsilon$ which is a ratio between volume of voids and total volume of the medium.
2. PPI (Pores Per Inch) which is the number of pores in one linear inch.

Unfortunately while reading some papers about the relationship between volumetric heat transfer coefficient and porosity in metal foams, some authors describe the range of porosity they used with $\epsilon$ while the others use PPI density, so is there any relation to connect both terms?

There is a similar question on cfd-online and someone answered with this formula but I am not sure where it came from or if it's even correct:

$$\text{Pore density (ppcm)}=\frac{300}{D_{particle}}\sqrt{\frac{(1-\epsilon)^2}{\pi\epsilon}}$$

The answer is system specific. That is, unless you know something else about the structure of your porous material, there is no clear relation between pore density and pore size. A simplified 2D example may look like this (where the circles are pores):

In both of these samples the pore density in ppi is the same, but the porosity is different. They are approximately 4 ppi samples but in the one on the left the porosity is about 0.5 and the one on the right the porosity is much less than that.

I have confirmed this through a quick search of literature on foams like the one you describe. This one (behind a paywall) for example has a table where they list their samples and some of them have the same ppi but different porosities. In their works they describe their porous material as being made of pores with diameters $d_p$ and fibers of diameter $d_f$. Given that additional information, I guess someone could develop a relationship between pore density and pore size.

• +1 Seems logical, I actually was trying to find a generic method (system independent) to connect both designations but I see how that might not work now. – Algo Jun 3 '16 at 14:28
• And giving it some thoughts in the light of your answer, why not use porosity solely if PPI designation could be somehow "ambiguous" when it comes to describe the porous structure? – Algo Jun 3 '16 at 14:30
• My guess is that some researchers don't have the means (or the manufacturer information) to measure the porosity so they just use the PPI. – Salomon Turgman Jun 3 '16 at 16:24
• @sturgman I would actually think that it would be the other way around. Porosity sounds easier to measure (i.e. it is basically density). PPI could involve microscopes and counting. Or does PPI imply pores that go completely through a sample? Basically open-cell versus closed-cell foam? – hazzey Jun 3 '16 at 19:40
• @hazzey Hmm. I was thinking of small pores where you would use BET type analysis to measure porosity. But you might be right. In some cases porosity might be easier to measure. – Salomon Turgman Jun 3 '16 at 21:47