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Attiogbe Spacing Equation

Recently, Attiogbe proposed a spacing equation which estimates the "mean spacing of air voids" in concrete [5]. From the author's figures, it appears as though the Attiogbe spacing equation attempts to estimate one half the minimum surface-surface spacing among neighboring air voids. An accurate numerical test of the equation is complicated by the exact definition of what the author's spacing equation attempts to quantify. Figure 1 of Ref. [5] depicts the "spacings" considered. In that figure, the author has chosen the nearest three voids as neighbors. The author should have included the other six voids that are "visible" to the central void since, by the author's definition, " is defined by considering only the distances, between adjacent air voids, which are entirely occupied by paste"[5].

A definitive numerical test of the Attiogbe[5] spacing equation is complicated further by the author's ambiguous definitions of certain mathematical quantities. The initial spacing equation proposed by Attiogbe, "valid for all values of p/A," [5] was

(To avoid confusion with the other spacing equations presented here, the variable t has been substituted for in the author's original equation.) Upon noting that this equation has peculiar properties for some values of p/A[19], Attiogbe has since been using the equation

in more recent publications [8]. (The variable G replaces the author's variable F to avoid confusion with the Philleo spacing factor.) The author states that, "[G]... is the fraction of the total paste volume within the distances of [t] from the edges of the air voids ... In this regard, [G] is equivalent to the probability factor defined in Philleo's 'protected paste volume concept' " [5]. In its simplest form, it is represented by the equation

However, the quantity G depends upon the air void radius distribution. Fortunately, Attiogbe has recently given an explicit equation for G for an air void diameter distribution based upon the gamma function [20]

The parameters a and b can be related to the mean diameter and the variance of the distribution σ 2:

For any parameters (a,b), the equation for G is [21]

This result will be useful for the air void radii distributions used in this experiment. Additionally, since G is an estimate of the fraction of paste within t of an air void, it will be compared to measured values.

Next: Mean Free Path Up: Spacing Equations Previous: Philleo Spacing Equation