The Beta Distribution

馃帰 1. The Math of Liberation: The Beta Distribution

Engineers love curves. In mineral processing, our favorite is the Beta Distribution.

馃 Why Beta?

Because the Beta curve can take the shape of a U (like a bowl).

A well-ground ore usually has:

  • A lot of pure waste (left side)

  • A lot of pure mineral (right side)

  • Very little mixed particles (middle)

Perfect U-shape → Perfect for liberation modeling.


Beta curve = A U-shaped graph telling us how much of the ore is free mineral, free waste, or mixed junk.

馃摝 The Four Parameters You Actually Need

1) Average Grade

“How much metal is there in the rock overall?”

Like:
If a classroom has 20% toppers → average grade = 20%.

2) Variance

“How mixed is everything?”

Low variance → uniform
High variance → some rich particles, some poor particles

3) L₀ = Liberated Gangue

Fraction of pure waste particles.
(They are fully useless but easy to remove.)

4) L₁ = Liberated Mineral

Fraction of pure valuable particles.
(The particles we actually want!)

Memory trick:

L₀ = useless
L₁ = useful

Everything between L₀ and L₁ = middlings (mixed particles).

馃椇️ 2. The Andrews–Mika Diagram: The Map of What’s Possible

This is the most important diagram in liberation theory.

Think of it as Google Maps for particle breakage.

馃Л What does it tell you?

It predicts all the possible particles you can create when you break a rock of known size and grade.

馃彅️ Example

Your rock is:

  • Size: 10 mm

  • Grade: 20% gold

You smash it into dust.

Question:
Can you get a 5 mm pure gold nugget from this breakage?

Answer:
HELL NO.
You can’t create gold that wasn’t there before.

The Attainable Region

This is the “legal zone” in the Andrews–Mika diagram.

It represents what is physically possible when you break an ore.

The Law of Conservation

You cannot produce particles richer than the richest mineral grain inside the parent rock.

The diagram creates mathematical limits around:

  • Possible grades of daughter particles

  • Possible sizes

  • Possible liberation states

Definition:

Andrews–Mika = The “Don’t dream unnecessarily” diagram.

3. Breaking Bad: Non-Random Fracture

Usually, we assume rocks break randomly (like when you drop a glass bowl).

But sometimes… rocks act “smart.”

 Preferential Breakage

Some minerals break more easily at:

  • Grain boundaries

  • Weak interfaces between valuable and waste minerals

This means:

 Result

The rock naturally splits right along the edge of the valuable mineral.

 Benefit

  • Higher liberation

  • Less grinding needed

  • Lower energy cost

  • Better separation performance

Explanation:

Rock sees gold… and decides to break around it.

4. Summary Checklist 

✔️ Liberation

Breaking rocks until minerals are freed from gangue.

✔️ Grain Size Rule

If grain is 50 microns → grind smaller than 50 microns.

✔️ Stereology

Using 2D microscope images to guess 3D mineral shapes.
(Like judging a cake by one slice.)

✔️ Beta Distribution

The U-shaped mathematical curve describing how much material is:

  • Pure waste (L₀)

  • Pure mineral (L₁)

  • Mixed (middle)

✔️ Andrews–Mika Diagram

A map of all possible particles you can create during breakage.
Shows the “legal” zone based on mass and grade constraints.




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