Chapter 2: Particle Populations & Distribution Functions
Chapter 2: Particle Populations & Distribution Functions
1. Introduction: The Problem of "Too Many Rocks"
In mineral processing (crushing, grinding, flotation), we deal with billions of particles at once. It is physically impossible to measure the size and mineral content of every single rock in a grinding mill.
Instead of tracking individuals, we use Statistics. We treat the rocks like a population of people. We don't ask, "How tall is John?"; we ask, "What percentage of the population is between 5 and 6 feet tall?"
Key Terms
External Coordinates: Where the particle is located physically in the machine (x, y, z).
Internal Coordinates: What the particle is made of and what it looks like. The two most important internal coordinates are:
Size: Determines how it breaks.
Composition (Grade): Determines how we separate it (e.g., valuable copper vs. waste rock).
Real-World Example: Imagine a bag of mixed coins. You don't describe the bag by listing every single coin. You describe it by saying: "It is 50% quarters, 30% dimes, and 20% pennies." In mining, "Quarters" might be large rocks, and "Pennies" might be sand.
2. Particle Size Distribution Functions
Since rocks have irregular shapes (they aren't perfect spheres), we define size by sieving. The "size" of a particle is simply the smallest square mesh opening it can fall through.
A. The Cumulative Distribution Function: P(d)
This is the most common curve used in engineering.
Definition: P(d) tells you the mass fraction (percentage) of the sample that is smaller than or equal to size d.
The Rules:
P(0) = 0: You cannot have rocks smaller than zero.
P(infinity) = 1: 100% of your rocks are smaller than infinity.
Monotonically Increasing: The curve never goes down. As the size gets bigger, the amount of material passing that size must stay the same or increase.
B. The Discrete Density Function: p(i) (The "Bin" Method)
In the lab, we don't get a smooth curve; we get specific weights on specific sieves.
Definition: p(i) is the fraction of material trapped between two sieves (a top screen and a bottom screen).
Formula: Mass in Bin = (Mass Passing Top Screen) - (Mass Passing Bottom Screen).
C. Representative Size
When we do math, we can't use a "range" like 10mm to 20mm. We need a single number to represent that bin.
Definition: The single size used to represent all particles in a specific sieve class.
Calculation: We usually use the Geometric Mean of the top and bottom screen sizes.
Formula: Representative Size = Square Root of (Size Top x Size Bottom).
Interesting Fact: Why do standard laboratory sieves usually increase in size by a factor of roughly 1.41 (square root of 2)? Because experience shows this spacing puts roughly equal amounts of mass on every sieve, making the data easier to analyze.
3. Empirical Distribution Models
Engineers love straight lines because they are easy to read. We use mathematical equations to describe rock piles so we can turn curved data into straight lines on special graph paper.
A. The Rosin-Rammler Distribution
Use: Very common for broken rock (crushed or ground ore).
What it means: It uses two numbers to describe the pile: one for how coarse the sample is, and one for the "spread" (wide mix of sizes vs. uniform size).
B. The Lognormal Distribution
Use: Often seen in nature or after material has been broken many, many times.
What it means: It looks like a Bell Curve (Gaussian), but plotted on a logarithmic scale.
C. The Gaudin-Meloy Distribution
Use: A "Truncated" distribution.
What it means: Sometimes a rock pile has a strict maximum size (e.g., nothing larger than the crusher gap can get through). This function forces the curve to stop at a maximum size.
Interesting Fact: In 1941, the mathematician A.N. Kolmogorov proved that if you break a rock randomly, then break the pieces randomly, and keep doing that forever, the particle sizes will mathematically turn into a Lognormal Distribution.
4. Mineral Composition and Liberation
In mining, size isn't everything. We need to separate value (Mineral) from waste (Gangue).
A. Mineral Grade (g)
Definition: The mass fraction of a specific particle that is valuable mineral.
Liberation Classes:
Grade = 0 (Liberated Gangue): Pure waste rock.
Grade = 1 (Liberated Mineral): Pure valuable metal.
Grade between 0 and 1 (Middlings/Locked): A particle that is part value, part waste.
B. The Goal of Comminution (Crushing)
The goal isn't just to make rocks smaller; it is to Liberate them. We break "Middlings" to try and create free particles (Grade=1) that we can sell, and free waste (Grade=0) that we can throw away.
Real-World Example: Think of a Chocolate Chip Cookie.
The Cookie: The Ore.
The Dough: The Waste (Gangue).
The Chips: The Value (Mineral).
Liberation: If you smash the cookie, you get some pure crumbs (waste), some pure chocolate chips (value), and some chunks that are dough with half a chip stuck to it (middlings).
5. Advanced Distributions: Joint & Conditional
A. Joint Distribution
Real particles possess a size AND a grade at the same time.
Definition: Describes the population using a 2D Grid.
Example: "What percentage of the pile is BIG (Size > 10mm) AND RICH (Grade > 50%)?".
B. Conditional Distribution (The "Washability" Curve)
This answers "What if?" questions.
Definition: The grade distribution of a specific subset of the population.
Example: "If I only look at the small particles, what is their mineral grade?"
How to measure: We take a specific size fraction (from sieving) and perform a Heavy Liquid Separation (Sink-Float) test to see how much heavy mineral is inside that specific size.
6. The "Bank Account" of Rocks: Population Balance
This is the unified framework used to build computer simulators for crushers, mills, and flotation cells. It treats the machine like a bank account for rocks.
The Equation: Accumulation = Input - Output + Birth - Death
Input: Ore entering the mill.
Output: Ore leaving the mill.
Death: When a large rock breaks, it "dies" (it disappears from the large size class).
Birth: When that large rock breaks, fragments appear in the smaller size classes. They are "born".
Conservation of Mass: Mass is never lost. If a 1kg rock "dies," exactly 1kg of smaller sand is "born".
7. Summary Checklist for Students
Distribution Functions: The statistical tool used to describe billions of particles.
Discrete Density: Grouping rocks into "bins" (sieves) to make calculation easy.
Representative Size: Using the geometric mean to represent a bin.
Composition: Tracking how much valuable mineral is inside each particle.
Population Balance: Tracking the flow of mass as particles break (birth/death) and move through the plant.
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