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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 *t _{G}* 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
*t _{G}* that
warrants further discussion.
Based upon Eqn. 10,
The quantity

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 >, < R ^{2} >*, and

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