# AMIT 135: Lesson 3 Particle Size Distribution

## Objectives

After completing this lesson students should be able to:

• Demonstrate understanding of various method involved in measuring particle size.
• Explain the importance of particle size analysis and its application.
• Differentiate between the different models used for predicting particle size.
• Explain the different convention followed in measuring particle size.

### Mineral Particle Sizes

• Particles within a given process stream vary in size and shape.
• Quantifying the amount of material within a given size fraction is often important for design and operational considerations.
• Particles that are either spherical or cubical are relatively easy to characterize.
• However, the majority of particles are often neither of these two shape types.
• Particle size characterization can be determined by:
• Microscopic Analysis
• Sieving
• Sedimentation (Stokes Diameter)
• Optical (Laser deflection or reflection)

### Mean Particle Sizes

• The mean particle size of a distribution is typically measured by either the arithmetic or geometric mean of the maximum (dmax) and minimum (dmin) particle sizes.
• The arithmetic mean is most accurate for symmetric particles such as spheres or cubes.
$d{am}&space;=&space;\frac{d{max}&space;+&space;d{min}}{2}$
• The geometric mean is typically preferred when segregation of the particles into various fraction were achieved by passing particles through an opening having a given shape and area, e.g., screening.
$d{gm}&space;=&space;(dmax&space;*&space;dmin)^{\frac{1}{2}})&space;=&space;(d{1}d{2}d{3}...d{n})^{\frac{1}{N}}$N = number of particles or size fractions

### Particle Size Analysis

Sieve Opening Size (mm)Mean Particle Size (mm)Weight (#)Cumul. % RetainedCumul. % Finer
PassedRetained
(1*2.5)1.01.195.05.0100.0
1.00.60.7710.015.095.0
0.60.30.4225.040.085.0
0.30.150.2140.080.060.0
0.15pan0.0120.0100.020.0
Total100.00
• The top size was estimated to be the square root of two times the top sieve size or 1.4mm.
• There is 5% retained on the 1mm screen.
• If the total feed was directed to the 0.3mm screen, 40% of the feed would be retained on the screen.
• 95% of the feed is finer than 1mm and 20% is finer than 0.15mm.
• The mean size of the material passing the 0.15mm screen can be estimated assuming the bottom size is 1 micron.

### Particle Sizes Distribution Models

• There is a common need to determine the amount of material in the feed at a given particle size.
• The desired particle size may not have been included in the original particle size analysis.
• The two common models used include:
1. Gates-Gaudin-Schuhmann Model (GGS)
2. Rosin-Rammler Model (RR)
• The RR model is typically satisfactory for coarse distributions.
• The GGS model is generally considered more precise for fine particle size distributions.
• It may be needed to perform model fits over more than 2 particle size rangers.

#### Gates-Gaudin-Schumann Model

The GGS model predicts the cumulative percent passing distribution:

$Y&space;=&space;100(\tfrac{x}{k})^{m}$

Y= cumulative percent passing

x= particle size

k= size parameter

m =distribution parameter

The values of k and m can be determined by linear regression:

log y = m log x + k

#### Rosin-Rammler Model (RR)

• The Rosin-Rammler model is typically used to predict the % retained.
• The modified equation to predict the % finer is:$Y&space;=&space;100&space;-&space;100exp&space;\left&space;[&space;-\left&space;(&space;\frac{x}{R}&space;\right&space;)^{b}&space;\right&space;]$

R= size parameter

b= distribution parameter

• There is special graph paper available to help determine the correct values of R and b over any size range.

### Volume Mean Diameter

The mean particle size by volume is important when dealing with topics such as material transport, storage and hindered particle settling velocities.

The volume mean diameter, $d_{3}^{v}$ , can be quantified by:

$d_v&space;=&space;\left&space;(\frac{\sum_{i=1}{M_i}}{\sum_{i=1}\frac{M_i}{d^3}}&space;\right&space;)^\frac{1}{3}&space;=&space;\left&space;(&space;\frac{1}{\sum_{i=1}\frac{M_i}{d^3}}&space;\right&space;)^\frac{1}{3}$

Where $M_{i}$ is the mass within size fraction i.

Mean Size Fraction Diameter (mm)dp3Mass, Mi (%)Mi/d3
0.80.5121019.53
0.60.21640185.19
0.30.027301111.11
0.1250.0022010000
Total10011315.83

$d_v&space;=&space;\left&space;(&space;\frac{100}{11315.83}&space;\right&space;)^\frac{1}{3}&space;=&space;0.206mm$

### Surface Mean Diameter

• The surface mean diameter is important when considering surface coatings with chemicals, particle agglomeration and dewatering.
• The volume mean diameter, ds , can be quantified by:$d_{s}&space;=&space;\left&space;(&space;\frac{particle&space;mass&space;per&space;diameter}{particle&space;mass&space;per&space;volume}&space;\right&space;)^{\frac{1}{2}}&space;=&space;\left&space;(&space;\frac{\Sigma_{i-1}&space;\frac{M_{i}}{d_{i}}}{\Sigma_{i-1}&space;\frac{M_{i}}{d^{3}}}&space;\right&space;)^{\frac{1}{2}}$
where Mi is the mass within the size fraction i.
Mean Size Fraction Diameter (mm) $\dpi{200}&space;d_{3}^{p}$ Mass, Mi (%) $\dpi{200}&space;\frac{M_{i}}{d_{3}^{i}}$ $\dpi{200}&space;\frac{M_i}{d_i}$
0.8 0.512 10 20 13
0.6 0.216 40 185 67
0.3 0.027 30 1111 100
0.125 0.002 20 10000 160
Total 100 11315 340

$d_{s}&space;=&space;\left&space;(&space;\frac{340}{11316}&space;\right&space;)^{\frac{1}{2}}&space;=&space;0.173$

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