The Basics of Particle Size
Part 1: The Basics of Particle Size (Syllabus 2.1 – 2.4)
2.1 Introduction: The "Too Many Rocks" Problem
In mineral processing (crushing, grinding, flotation), we deal with billions of particles at any given second. It is physically impossible to measure the diameter and weight of every single rock in a 100-ton truckload.
To solve this, we stop looking at individuals and start looking at the Population.
Analogy: Think of a Census. The government doesn't list every citizen's name on the news; they say "20% of the population is under age 18."
In Mining: We don't list every rock; we say "20% of the ore is smaller than 1mm."
We use Distribution Functions to describe the crowd mathematically.
2.2 Distribution Functions (The Cumulative Curve)
Because particles are irregular shapes (not perfect spheres), we define "size" by sieving. If a rock passes through a square hole of width d, it is considered "smaller than d."
The Cumulative Distribution Function: P(d)
This is the most common graph used in engineering. It answers the simple question: "What fraction of the total mass is smaller than size d?"
The 3 Golden Rules of P(d):
P(0) = 0: You cannot have rocks smaller than zero.
P(infinity) = 1: 100% of your rocks are smaller than infinity (all rocks are accounted for).
Monotonically Increasing: The curve never goes down. As you look at larger and larger sizes, the amount of material passing that size must stay the same or increase.
Empirical Models (Fitting the Curve)
Engineers use math equations to draw these curves smoothly. The three most common models are:
Rosin-Rammler: Used for broken/crushed rock. It uses two numbers: one for "coarseness" and one for the "spread" of sizes.
Lognormal: Used for materials that have been broken repeatedly (like in nature or ball mills). It looks like a Bell Curve plotted on a specific scale.
Gaudin-Meloy: Used when there is a strict maximum size limit (e.g., nothing bigger than the crusher gap can get through).
2.3 Distribution Density Function (The Continuous Curve)
While the Cumulative Function P(d) is a running total (summing up), the Density Function p(d) looks at a specific instant or point.
Definition: It is the slope of the cumulative curve.
Physical Meaning: It tells you the frequency (or amount) of particles at exactly size d.
Visual: If you plot this, it looks like a hill or a bell curve. The total area under the hill must always equal 1 (representing 100% of the mass).
2.4 Discrete Density & Representative Size (The "Bin" Method)
In the real world (and in computer simulations), we cannot handle smooth, continuous curves. We chop the size range into specific Classes or Bins (like sieves in a lab).
The Discrete Density: p(i)
Definition: The fraction of material trapped between two specific sieves.
Calculation: Mass in Bin = (Mass Passing Top Screen) - (Mass Passing Bottom Screen)
This creates a histogram (a bar chart) of the rock sizes.
Representative Size (d_pi)
When we do calculations, we can't use a range like "10mm to 20mm." We need a single number to represent all the rocks in that bin.
The Problem: Do we use 10? 20? 15?
The Solution: We use the Geometric Mean.
Formula: Representative Size = Square Root of (Top Size x Bottom Size)
Important Note: This method works best if the sieves are close together. Standard lab sieves usually increase by a factor of roughly 1.41 (the square root of 2) to keep the math accurate.
2.5 Distributions Based on Particle Composition
In mining, size isn't the only thing that matters. A rock can be the perfect size, but if it is just waste rock (gangue), it is useless. We need to measure the Mineral Grade.
Grade (g): This is the fraction of the particle that is valuable metal (like copper or gold).
The Three Types of Particles:
Liberated Mineral (g = 1): Pure treasure. This is what we want.
Liberated Gangue (g = 0): Pure waste. We want to throw this away.
Middlings (0 < g < 1): Locked particles. A rock that is half treasure, half waste. We usually have to crush these further to separate them.
Visual: Imagine a cookie. The dough is waste, the chips are value. If you break it, you get pure crumbs (waste), pure chips (value), and chunks of cookie with chips stuck in them (middlings).
2.6 Joint Distribution Functions (The Grid)
Real particles have Size AND Grade at the same time. We can't just look at one.
The Concept: Imagine a giant grid or checkerboard.
Rows: Size Classes (Big, Medium, Small).
Columns: Grade Classes (High Grade, Medium Grade, Waste).
Definition: The Joint Distribution tells you the mass fraction of material that falls into a specific box on this grid.
Example: "5% of the total mass is Big Rock (Size Class 1) that is also High Grade (Grade Class 1)."
2.7 Conditional Distributions (The "What If" Curves)
This answers specific questions by locking one variable.
Question: "If I look only at the sand-sized particles, what is their grade?"
Real World Application: This is called a Washability Curve.
How it works:
You take one size fraction (e.g., stones between 10mm and 20mm).
You drop them into "Heavy Liquids" of different densities.
Heavy rocks sink (usually the metal), light rocks float (usually the waste).
This tells you how easy it will be to separate the metal from the waste at that specific size.
2.8 Independence
Definition: Two properties are "independent" if changing one does not change the other.
In Mining: Size and Grade are almost NEVER independent. Usually, as you crush rocks smaller, the grade distribution changes (you get more liberated particles).
2.9 Distributions by Number
Usually, we measure mass (weight in tons). But sometimes, we need to count the number of particles.
The Problem: One giant boulder weighs the same as millions of grains of sand.
The Conversion: If you know the density and the volume (size cubed) of the particles, you can convert Mass Distributions into Number Distributions mathematically.
Part 3: Modeling the Machine (Syllabus 2.10 – 2.14)
2.10 Internal vs. External Coordinates
To simulate a machine effectively, we categorize the properties of the rock:
External Coordinates (Where is it?):
The physical location of the particle inside the tank or pipe (x, y, z).
Internal Coordinates (What is it?):
The inherent traits of the rock itself: Size, Mineral Grade, Hardness, Shape.
These travels with the particle.
2.11 Particle Properties
We use the internal coordinates to calculate other things.
Example: If we know the Mineral Composition (how much gold vs. quartz), we can calculate the Particle Density.
Gold is heavy. Quartz is light.
A particle with 50% gold and 50% quartz will have a density exactly in the middle.
2.12 The Population Balance Method
This is the "Grand Theory" of mineral processing simulation. It treats the machine (crusher, mill, flotation cell) like a bank account for rocks.
The Logic: We don't track a single rock. We track the amount of rock in every "bin" (Size/Grade class).
2.13 The Fundamental Equation
This equation accounts for everything happening in the machine.
Accumulation = Input - Output + Birth - Death
Input: Rocks entering the machine (Feed).
Output: Rocks leaving the machine (Product).
Death: A particle leaves its current bin.
Example: A large rock gets smashed. It "dies" from the Large Bin.
Birth: A particle appears in a new bin.
Example: The fragments of that smashed rock appear in the Small Bin. They are "born."
Conservation of Mass: You cannot destroy matter. The mass that "dies" from the large bin must equal the mass that is "born" in the smaller bins.
2.14 General Equation for Comminution (Crushing/Grinding)
For a crushing machine, the equation simplifies because rocks only get smaller.
Selection Function (Rate of Breakage): How likely is a rock of Size X to break?
Big rocks usually break faster than small rocks.
Breakage Function (Distribution of Fragments): When a rock of Size X breaks, what pieces does it create?
Does it shatter into dust? Or does it split into two halves?
The Simulation Goal: If we know the Input, the Selection Function (probability of breaking), and the Breakage Function (what pieces are made), the computer can predict exactly what the Output (product) will look like.
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