Types of Particle Size Distribution
When releasing particles, it is possible to specify a list of particle diameters or to sample the particle diameters from a distribution function. The built-in distribution functions include normal, lognormal, and uniform distributions.
Normal Distribution
For a normal distribution, the probability distribution function is
where
•
dp (SI unit: m) is the particle diameter,
•
μ (SI unit: m) is the average particle diameter, and
•
σ (SI unit: m) is the standard deviation of the particle diameter.
In this context, the average is an arithmetic mean over released particles,
(5-31)
The standard deviation is
(5-32)
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.
Lognormal Distribution
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.
Figure 5-9: Probability distribution function of particle diameter for three different values of the geometric standard deviation.
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 Sauter mean diameter is the diameter of a particle that has the same volume to surface area ratio as the population of particles as a whole,
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.
 
Table 5-4: Definitions of various average particle diameters, and their coefficients for the Hatch-Choate Equations.
Quantity
Expression
Coefficient
Count median diameter
50% of particles have diameter greater than the CMD.
0
Count mean diameter
0.5
Diameter of average surface area
1
Length median diameter
The sum of the diameters of particles larger than the LMD is 50% of the sum of all diameters.
1
Diameter of average mass
1.5
Length mean diameter
1.5
Surface median diameter
The total surface area of particles larger than the SMD is 50% of the total surface area.
2
Sauter mean diameter
(surface area mean diameter)
2.5
Mass median diameter
The amount of mass in particles larger than the MMD is 50% of the total mass.
3
Mass mean diameter
3.5