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A computer program can serve as an effective means to evaluate the mathematical performance of various spacing equations. The computer allows one to monitor both the size and location of every sphere in the system, for any air void radii distribution. This computerized approach helps to assure that the successful freeze-thaw prediction performance of an equation is due, in part, to the successful geometrical performance of the equation. Without this, one has no basis for correlating freeze-thaw performance with the prediction of an equation.
Four air void spacing equations were considered: Powers, Philleo, Pleau and Pigeon, and Attiogbe. In addition, an equation by Lu and Torquato which also estimates the same spacing characteristics was considered. For the lognormally distributed radii used, the Powers equation is within a factor of 1.5 of the 95 th percentile of the fraction of paste within an air void surface. The Philleo equation estimated the 50 th and 95 th percentiles of the paste-void spacing distribution fairly well. The Pleau and Pigeon equation also estimated these percentiles, but their equation did not perform as well as the Philleo equation for lognormally distributed sphere radii. The original Attiogbe equation is approximately a factor of two less than the paste mean free path and an order of magnitude greater than the 95 th percentile of the void-void spacing distribution. The Attiogbe equation for tG estimates an increasing spacing between monosized air voids as the number remains fixed and the diameter increases.
The Lu and Torquato equations performed quite well for both paste-void and void-void spacing distributions. Not only can their equations predict arbitrary statistics of both the paste-void and void-void spacing distributions, they also accurately predict the average void-void spacing as a function of void radius.
Due to the accuracy of the Lu and Torquato equation, it would appear as though additional spacing equations are not needed. The Powers equation can be tested further for various air void radii distributions and compared to the Lu and Torquato equation. If the Powers equation fails to consistently predict the same percentile of the paste-void spacing distribution, it could be replaced with either the Lu and Torquato equation or a simplified approximation. Regardless, as the material properties of concrete continue to change, effort must also address establishing an appropriate limit of allowable spacing in concrete.
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