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These results suggest that the Powers spacing factor approximates some large percentile of the paste-void spacing distribution, as it was intended. However, the exact percentile is unknown, and appears to vary with the air void radii distribution.

The Philleo equation performed much better for lognormally distributed sphere radii than for monosized sphere radii. It is not clear why this should be so. Philleo did not consider any air void radii distribution in the development of his equation.

Although the Pleau and Pigeon equation works well at very low air contents, the performance of the equation worsens with increasing air content, especially for the lognormally distributed sphere radii. This is a consequence of using the Hertz distribution for h(x). It is interesting to note that, for lognormally distributed air void radii, although the Pleau and Pigeon equation is a function of the sphere radii distribution, it does not perform as well as the Philleo equation, which is independent of the sphere radii distribution.

It should be mentioned that Pleau and Pigeon have pointed out that the deficiencies in their equation may be an advantage in real concretes contained entrained air voids. When the fact that K'(0) represents the air content of overlapping spheres was pointed out in a discussion of their work [23], Pleau and Pigeon argued that, "overlapping of air bubbles is frequently seen during the ASTM C 457 microscopical examination, especially for concretes having high air contents." [23]. Further research is needed to determine the extent of this effect, and its subsequent impact on estimates of the paste-void proximity distribution.

The performance of the Attiogbe equations in estimating percentiles of the void-void spacing distribution was quite poor. The original equation t is completely disconnected from the void-void spacing distribution, which is expected since it is proportional to the mean free path. The equation tG has unphysical behavior for monosized spheres with constant number density and increasing void radius. For lognormally distributed sphere radii, it is clear that neither equation has any relevance to any reported statistic of the void-void spacing distribution.

There is one quantitative aspect of the equation tG that warrants further discussion. Based upon Eqn. 10, The quantity tG has a maximum value of 16 / α as the number density n of air voids approaches zero, regardless of the air void radius distribution. This suggests that there is a finite distance between air voids when there are virtually no air voids present, which is completely unphysical.

The performance of the Lu and Torquato equation is, by far, the most accurate estimate for every statistic considered. Not only does it predict these statistics well, it also predicts the average void-void spacing as a function of radius for polydispersed sphere radii. It appears as though the Lu and Torquato equation is accurate to the level of precision required for investigations of air void spacing. These results also suggest that, at the air volume fractions investigated here, an air void distribution approximated by a collection of parked spheres has very similar spatial statistics to an equilibrium distribution of spheres, which has relevance to numerical tests of air void equations. It is also interesting to note that the Lu and Torquato equations do not require information about the entire air void radii distribution. Rather, only the values < R >, < R2 >, and
< R3 > are needed. This has significant importance for stereological practices [15].

Next: Alternative Attiogbe Equation Up: Main Previous: Paste-Void Probability Density