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# Appendix B: Using a sieve analysis to compute statistical quantities needed

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 have all the same density, then mass fractions are the same as volume fractions.

Now, in the ITZ volume formulas, averages appear of powers of the aggregate radii, averaged over the number distribution density of the aggregates. Below we show how this can be carried out using the sieve analysis, followed by formulas for performing volume averages using the sieve analysis, which is necessary to be able to evaluate the D-EMT formulas for a given aggregate particle size distribution.

In order to carry out these averages, we need to make an assumption 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. We show the analysis for both assumptions, although in all the simulation work, the former assumption was used. We could, of course, also assume 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 to assume some kind of distribution within the sieve.

Next: Assumption 1: Uniform Up: Main Previous: Appendix A: Discussion