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### Attiogbe Equation

Fortunately, Eqn. 13 can be used to determine the correct value of G to use for monosized spheres. The diameter distribution of Eqn. 11 proposed by Attiogbe can approximate a monosized distribution when the variance goes to zero. From Eqn. 12, this will be true when the parameter a approaches infinity, and b approaches zero. In the limit that a goes to infinity, the value of G calculated from Eqn. 13 is

Note that this value differs very little from the simpler form of Eqn. 10. This small variation is in complete agreement with the results and conclusions of Attiogbe [21]. The value of G in Eqn. 39 is that used for the monosized sphere experiment.

The performance of both Attiogbe spacing equations are shown in Table 3 and Table 4. Table 3 shows the estimates of the 50 th and 95 th percentiles of the void-void spacing distribution. Table 4 shows the estimates of the average minimum spacing between voids. It appears as though neither Attiogbe equation can accurately predict the 50 th percentile, the 95 th percentile, or the mean of the void-void spacing distribution. However, as expected, the Attiogbe equation t is consistently one half the mean free path ().

Table 5 shows the performance of G in predicting the fraction of paste within either t or tG of an air void. Since the Lu and Torquato equation for EV performed so well for monosized spheres, it is used to represent the correct value. From the results, it would appear as though G is not an accurate estimator of the volume of paste within either t or tG of an air void for monosized spheres.

There are aspects of the equation tG in the constant number density experiment that warrant attention. First, in Table 3 the estimated spacing using tG at zero air is 0.0000, which is unreasonable. If, as Attiogbe has defined his spacing, this is the average minimum spacing, all the spheres must be touching one another since there can be no negative distances between voids. However, as described previously, at zero air content in this experiment the voids are simply point particles, and they cannot be touching one another. Second, in the same table the estimated spacing tG increases with increasing air content, and then decreases sharply at paste air content of about 12%, which corresponds to a concrete air content of about 4%. An increase in an estimate of the average minimum spacing between bubbles, with increasing air content, is unphysical for identical air voids.

Next: Lu and Torquato Up: Mono-sized Spheres Previous: Pleau and Pigeon