Because of the way initial values of auxiliary dependent variables are sampled in particle release features, Equation 5-31 will be correct to an extremely high degree of precision even if the number of particles is relatively small. In contrast, Equation 5-32 will only approximately be respected, with the standard deviation over released particles approaching the specified value σ as the number of particles N is increased.

A lognormal distribution of released particle diameters follows the probability distribution function (Ref. 42)

where CMD (SI unit: m) is the count median diameter and GSD (dimensionless) is the geometric standard deviation. In general GSD should be greater than 1. By definition, half of the released particles have diameter greater than the count median diameter.

The relationship between the geometric standard deviation GSD and the standard deviation from Equation 5-32, simply labeled σ, is

For a lognormal distribution, the mean diameter μ, count median diameter CMD, and Sauter mean diameter dsm are related by

The equations relating CMD, μ, and dsm are special cases of a general class of conversion equations known as the Hatch–Choate Equations, following a general form

where dA is some type of average diameter yet to be specified and b is a dimensionless constant. The values of b for some typical average quantities are given in Table 5-4.