Next: Assumption 2: Uniform Up: Appendix B: Using Previous: Appendix B: Using

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 (V_i+1, V_i) will be equal to c_i. If N is the total number of aggregate particles used per the total concrete volume V_TOT, so that ρ = N/V_TOT, V_agg is the total aggregate volume, c_agg = V_agg / V_TOT, 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