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## Assumption 1: Uniform distribution by volume

In this case, the fraction of the total aggregate volume represented by particles with volumes in the range (V, V + dV), contained in the ith sieve, is given by

so that the integral over the interval (Vi+1, Vi) will be equal to ci. If N is the total number of aggregate particles used per the total concrete volume VTOT, so that ρ = N/VTOT, Vagg is the total aggregate volume, cagg = Vagg / VTOT, then the fraction of the total number of aggregate particles with volumes in the range (V, V + dV), contained in the ith sieve, is given by

where V is the volume of a particle in this range. If we now convert to radius, using V = 4πr3 / 3 and dV = 4πr2dr, the equivalent expression in terms of the particle radius is

Integrating over each sieve's endpoints and summing over each sieve must give 1 for this expression:

This normalization determines the value of ρ:

Therefore, the average of Rn over the particle number density, as shown in eq. (19), is

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