D-values and Undersize Fractions for AMUSIG

The AMUSIG model for dispersed Eulerian multiphase flow simulations provides the D-values and Undersize Fractions post-processing options to analyze the particle size distribution by specific size groups.

Sieve analysis is a common method that is used to determine the particle size distribution of granular or powdered materials [518]. This analysis involves passing a sample of material through a series of sieves with various aperture sizes, and measuring the amount of material retained on each sieve.

The example plot below displays a typical particle size distribution function

F

, which represents the mass distribution of particles within a sample. For a sieve with aperture size

d

, the portion of the granular material that can pass through the sieve is:

Figure 1. EQUATION_DISPLAY

Φ = F (d)

(2292)

where

$Φ$ is the cumulative undersize fraction distribution, that is, the percentage (or fraction) of particles smaller than a given size.
$d$ is the cut diameter, that is, the percentage of material retained on each sieve plotted against the sieve aperture size.

To obtain the aperture size for a desired undersize fraction, the function

F

is inverted to define the cut diameter as:

Figure 2. EQUATION_DISPLAY

d = F^{- 1} (Φ)

(2293)

The cut diameter

d

also referred to as the D-value

D_{X}

, is calculated based on cumulative particle distribution data, where the X-axis represents the particle size and the Y-axis represents the mass percentage of particles in the sample.

The D-value characterizes the particle size at which a specific proportion of particles in a given sample are smaller. It is calculated by dividing the mass of the sample by its diameter. For example, $D_{50}$ value represents the particle size at which 50% of the particles in the sample (by mass) have a smaller diameter than $D_{50}$ while the remaining 50% have a larger diameter. Similarly, the $D_{10}$ value gives the particle size where 10% of the particle sample are smaller than that size.

The most common D-values are $D_{10}$ , $D_{50}$ , and $D_{90}$ . $D_{50}$ represents the median diameter. $\frac{D_{90} - D_{10}}{D_{50}}$ gives the range of the particle size distribution.

As D-values focus solely on the ratio of masses, it is not necessary to know the exact mass of the particles or the sample. The total mass is split depending on the choice of groups. For example, for 5 size-groups each group accounts for 20% of the total mass. When

D_{X}

does not coincide with any diameter of the groups it is estimated by a linear interpolation using two nearest size-groups.

Note	To resolve the smallest or biggest decile of the size distribution, more than 10 size-groups are required. The median diameter can be reliably calculated with 5 size-groups. The calculation of D10 and D90 values requires more CPU time.