Assumption 2: Uniform distribution by radius

In this case, the fraction of the aggregate volume represented by particles with radii in the range (r, r+dr), contained in the i'th sieve, is given by

$$p_i(r) dr = \frac{c_i dr}{(r_{i+1} - r_i)}$$ (17)

so that the integral of p_i(r)dr over the interval (r_i, r_i+1) will be equal to c_i. Similar to the previous case, the fraction of the total number of aggregate particles with radii in the range (r, r+dr), contained in the i'th sieve, is given by

$$n_i(r) dr = \frac{3 c_{agg} c_i dr}{\rho 4 \pi r^3 (r_{i+1} - r_i)}$$ (18)

where V = 4 r ³ / 3 is the volume of a (spherical) particle in this range. Eq. (18) must obey the normalization (14), implying that the value of $\rho$ is then

$$\rho = \sum_{i=1}^{M} \int _{r_i}^{r_{i+1}} \frac{3 c_{agg} c_i r^{-3} dr}{4 \pi (r_{i+1} - r_i)}$$ (19)

or

$$\rho = \sum_{i=1}^{M} \frac{3 c_{agg} c_i (r_i + r_{i+1})}{8 \pi (r_i r_{i+1})^2}$$ (20)

Therefore, the average of Rⁿ over the particle number density is

$$\langle R^n \rangle = \sum_{i=1}^{M} \frac{3 c_{agg} c_i}{4 \pi \rho (r_{i+1} - r_i)} \int _{r_i}^{r_{i+1}} r^{n - 3} dr$$ (21)