In order to make use of the analytical parts of the multi-scale model, one must be able to compute quantities like the average radius of the aggregates, weighted according to particle number statistics, not particle mass statistics. Sieve analyses are commonly measured in concrete technology, and so the procedure to use the results of a sieve analysis to compute these quantites is given below.
A typical sieve analysis of an aggregate can be expressed in terms of di, M, and ci, where ci is the fraction of the total volume of aggregate that has a diameter between di and di+1, di < di+1, and M is the total number of sieves used. The units of the particle diameters are millimeters. The sum of ci over the M sieves equals 1. A typical sieve analysis is expressed in terms of the mass fraction passing or retained by a certain sieve size, which can easily be converted to the form given here. If aggregates of different size all have the same density, then mass fractions are the same as volume fractions. Otherwise, conversion between mass fractions and volume fractions must be carried out.
In the ITZ volume formulas, eqs. (4)-(6), one finds powers of the aggregate radii, averaged over the number distribution density of the aggregates. The formulas below were derived to carry out this procedure using the sieve analysis. This is followed by formulas for performing volume averages, also using the sieve analysis, which are necessary to be able to evaluate the D-EMT formulas (<m>V ) for a given aggregate particle size distribution.
In order to carry out these averages, an assumption is needed as to how the aggregates are distributed within each sieve. That information is not given by a sieve analysis. Many assumptions are possible, but two that are easy to handle analytically, and are physically reasonable, are that either the aggregates are distributed, within a sieve, uniformly by volume or uniformly by diameter. The analysis is shown for both assumptions, although in all previous simulation work, the former assumption was used [6,51]. The assumption could also be made, of course, that all the aggregates in a sieve have the same radius, perhaps equal to the average of the endpoints of the sieve range, but it is more accurate and realistic to assume some kind of distribution within the sieve.