Averages over the volume distribution

For averages over the volume distribution of the particles, either assumption can again be used. The starting points are the same for each assumption. Assuming a uniform distribution by volume within a sieve of the particles, the volume average < ... >_V is

$$\langle f(r) \rangle_V = \sum_{i=1}^M \frac{3 c_i}{(r_{i+1}^3 - r_i^3)} \int_{r_i}^{r_{i+1}} r^2 f(r) dr$$ (22)

The equation for the assumption of a uniform distribution, within a sieve, of the particles by radius, is

$$\langle f(r) \rangle_V = \sum_{i=1}^M \frac{c_i}{(r_{i+1} - r_i)} \int_{r_i}^{r_{i+1}} f(r) dr$$ (23)